nLab geometry of physics -- homotopy types

Basic notions of homotopy theory

Traditionally, mathematics and physics have been founded on set theory, whose concept of sets is that of “bags of distinguishable points”.

But fundamental physics is governed by the gauge principle. This says that given any two “things”, such as two field histories $x$ and $y$, it is in general wrong to ask whether they are equal or not, instead one has to ask where there is a gauge transformation

$$x \stackrel{\gamma}{\longrightarrow} y$$

between them. In mathematics this is called a homotopy.

This principle applies also to gauge transformations/homotopies themselves, and thus leads to gauge-of-gauge transformations or homotopies of homotopies

and so on to ever higher gauge transformations or higher homotopies:

This shows that what $x$ an $y$ here are elements of is not really a set in the sense of set theory. Instead, such a collection of elements with higher gauge transformations/higher homotopies between them is called a homotopy type.

Hence the theory of homotopy types – homotopy theory – is much like set theory, but with the concept of gauge transformation/homotopy built right into its foundations. Homotopy theory is gauged mathematics.

A classical model for homotopy types are simply topological spaces: Their points represent the elements, the continuous paths between points represent the gauge transformations, and continuous deformations of paths represent higher gauge transformations. A central result of homotopy theory is the proof of the homotopy hypothesis, which says that under this identification homotopy types are equivalent to topological spaces viewed, in turn, up to “weak homotopy equivalence”.

In the special case of a homotopy type with a single element $x$, the gauge transformations necessarily go from $x$ to itself and hence form a group of symmetries of $x$.

This way homotopy theory subsumes group theory.

If there are higher order gauge-of-gauge transformations/homotopies of homotopies between these symmetry group-elements, then one speaks of 2-groups, 3-groups, … n-groups, and eventually of ∞-groups. When homotopy types are represented by topological spaces, then ∞-groups are represented by topological groups.

This way homotopy theory subsumes parts of topological group theory.

Since, generally, there is more than one element in a homotopy type, these are like “groups with several elements”, and as such they are called groupoids (Def. ).

If there are higher order gauge-of-gauge transformations/homotopies of homotopies between the transformations in such a groupoid, one speaks of 2-groupoids, 3-groupoids, … n-groupoids, and eventually of ∞-groupoids. The plain sets are recovered as the special case of 0-groupoids.

Due to the higher orders $n$ appearing here, mathematical structures based not on sets but on homotopy types are also called higher structures.

Hence homotopy types are equivalently ∞-groupoids. This perspective makes explicit that homotopy types are the unification of plain sets with the concept of gauge-symmetry groups.

An efficient way of handling ∞-groupoids is in their explicit guise as Kan complexes (Def. below); these are the non-abelian generalization of the chain complexes used in homological algebra. Indeed, chain homotopy is a special case of the general concept of homotopy, and hence homological algebra forms but a special abelian corner within homotopy theory. Conversely, homotopy theory may be understood as the non-abelian generalization of homological algebra.

Hence, in a self-reflective manner, there are many different but equivalent incarnations of homotopy theory. Below we discuss in turn:

• Topological homotopy theory

  ∞-groupoids modeled by topological spaces. This is the classical model of homotopy theory familiar from traditional point-set topology, such as covering space-theory.

• Simplicial homotopy theory.

  ∞-groupoids modeled on simplicial sets, whose fibrant objects are the Kan complexes. This simplicial homotopy theory is Quillen equivalent to topological homotopy theory (the “homotopy hypothesis”), which makes explicit that homotopy theory is not really about topological spaces, but about the ∞-groupoids that these represent.

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Ideally, abstract homotopy theory would simply be a complete replacement of set theory, obtained by removing the assumption of strict equality, relaxing it to gauge equivalence/homotopy. As such, abstract homotopy theory would be part and parcel of the foundations of mathematics themselves, not requiring any further discussion. This ideal perspective is the promise of homotopy type theory and may become full practical reality in the next decades.

Until then, abstract homotopy theory has to be formulated on top of the traditional foundations of mathematics provided by set theory, much like one may have to run a Linux emulator on a Windows machine, if one does happen to be stuck with the latter.

A very convenient and powerful such emulator for homotopy theory within set theory is model category theory, originally due to Quillen 67 and highly developed since. This we introduce here.

The idea is to consider ordinary categories (Def. ) but with the understanding that some of their morphisms

$$X \overset{f}{\longrightarrow} Y$$

should be homotopy equivalences (Def. ), namely similar to isomorphisms (Def. ), but not necessarily satisfying the two equations defining an actual isomorphism

$$f^{-1} \circ f \;=\; id_{X} \phantom{AAAA} f \circ f^{-1} \;=\; id_Y$$

but intended to satisfy this only with equality relaxed to gauge transformation/homotopy:

(1)$$f^{-1} \circ f \;\overset{gauge}{\Rightarrow}\; id_{X} \phantom{AAAA} f \circ f^{-1} \;\overset{gauge}{\Rightarrow}\; id_Y \,.$$

Such would-be homotopy equivalences are called weak equivalences (Def. below).

In principle, this information already defines a homotopy theory by a construction called simplicial localization, which turns weak equivalences into actual homotopy equivalences in a suitable way.

However, without further tools this construction is unwieldy. The extra structure of a model category (Def. below) on top of a category with weak equivalences provides a set of tools.

The idea here is to abstract (in Def. below) from the evident concepts in topological homotopy theory of left homotopy (Def. ) and right homotopy (Def. ) between continuous functions: These are provided by continuous functions out of a cylinder space $Cyl(X) = X \times [0,1]$ or into a path space $Path(X) = X^{[0,1]}$, respectively, where in both cases the interval space $[0,1]$ serves to parameterize the relevant gauge transformation/homotopy.

Now a little reflection shows (this was the seminal insight of Quillen 67) that what really matters in this construction of homotopies is that the path space factors the diagonal morphism from a space $X$ to its Cartesian product as

$$diag_X \;\colon\; X \underoverset{\text{weak equiv.}}{\text{ cofibration }}{\longrightarrow} Path(X) \overset{\text{ fibration }}{\longrightarrow} X \times X$$

while the cylinder serves to factor the codiagonal morphism as

$$codiag_X \;\colon\; X \sqcup X \overset{ \text{cofibration} }{\longrightarrow} Cyl(X) \underoverset{ \text{weak equiv} }{ \text{fibration} }{\longrightarrow} X$$

where in both cases “fibration” means something like well behaved surjection, while “cofibration” means something like satisfying the lifting property (Def. below) against fibrations that are also weak equivalences.

Such factorizations subject to lifting properties is what the definition of model category axiomatizes, in some generality. That this indeed provides a good toolbox for handling homotopy equivalences is shown by the Whitehead theorem in model categories (Lemma below), which exhibits all weak equivalences as actual homotopy equivalences after passage to “good representatives” of objects (fibrant/cofibrant resolutions, Def. below). Accordingly, the first theorem of model category theory (Quillen 67, I.1 theorem 1, reproduced as Theorem below), provides a tractable expression for the hom-sets modulo homotopy equivalence of the underlying category with weak equivalences in terms of actual morphisms out of cofibrant resolutions into fibrant resolutions (Lemma below).

This is then generally how model category-theory serves as a model for homotopy theory: All homotopy-theoretic constructions, such as that of long homotopy fiber sequences (Prop. below), are reflected via constructions of ordinary category theory but applied to suitably resolved objects.

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Literature (Dwyer-Spalinski 95)

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Definition

(model category)

A model category is

1. a category $\mathcal{C}$ (Def. ) with all limits and colimits (Def. );

2. three sub-classes $W, Fib, Cof \subset Mor(\mathcal{C})$ of its class of morphisms;

such that

1. the class $W$ makes $\mathcal{C}$ into a category with weak equivalences, def. ;

2. The pairs $(W \cap Cof\;,\; Fib)$ and $(Cap\;,\; W\cap Fib)$ are both weak factorization systems, def. .

One says:

• elements in $W$ are weak equivalences,

• elements in $Cof$ are cofibrations,

• elements in $Fib$ are fibrations,

• elements in $W\cap Cof$ are acyclic cofibrations,

• elements in $W \cap Fib$ are acyclic fibrations.

The form of def. is due to (Joyal, def. E.1.2). It implies various other conditions that (Quillen 67) demands explicitly, see prop. and prop. below.

We now dicuss the concept of weak factorization systems (Def. below) appearing in def. .

Factorization systems

Definition

(lift and extension)

Let $\mathcal{C}$ be any category. Given a diagram in $\mathcal{C}$ of the form

$$\array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow \\ B }$$

then an extension of the morphism $f$ along the morphism $p$ is a completion to a commuting diagram of the form

$$\array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ B } \,.$$

Dually, given a diagram of the form

$$\array{ && A \\ && \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y }$$

then a lift of $f$ through $p$ is a completion to a commuting diagram of the form

$$\array{ && A \\ &{}^{\mathllap{\tilde f}}\nearrow& \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y } \,.$$

Combining these cases: given a commuting square

$$\array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow && \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 }$$

then a lifting in the diagram is a completion to a commuting diagram of the form

$$\array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow &\nearrow& \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 } \,.$$

Given a sub-class of morphisms $K \subset Mor(\mathcal{C})$, then

• a morphism $p_r$ as above is said to have the right lifting property against $K$ or to be a $K$-injective morphism if in all square diagrams with $p_r$ on the right and any $p_l \in K$ on the left a lift exists.

dually:

• a morphism $p_l$ is said to have the left lifting property against $K$ or to be a $K$-projective morphism if in all square diagrams with $p_l$ on the left and any $p_r \in K$ on the left a lift exists.

Definition

(weak factorization systems)

A weak factorization system (WFS) on a category $\mathcal{C}$ is a pair $(Proj,Inj)$ of classes of morphisms of $\mathcal{C}$ such that

1. Every morphism $f \colon X\to Y$ of $\mathcal{C}$ may be factored as the composition of a morphism in $Proj$ followed by one in $Inj$

   $$f\;\colon\; X \overset{\in Proj}{\longrightarrow} Z \overset{\in Inj}{\longrightarrow} Y \,.$$

2. The classes are closed under having the lifting property, def. , against each other:

   1. $Proj$ is precisely the class of morphisms having the left lifting property against every morphisms in $Inj$;

   2. $Inj$ is precisely the class of morphisms having the right lifting property against every morphisms in $Proj$.

Definition

(functorial factorization)

For $\mathcal{C}$ a category, a functorial factorization of the morphisms in $\mathcal{C}$ is a functor

$$fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}$$

which is a section of the composition functor $d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.

Remark

In def. we are using the following standard notation, see at simplex category and at nerve of a category:

Write $[1] = \{0 \to 1\}$ and $[2] = \{0 \to 1 \to 2\}$ for the ordinal numbers, regarded as posets and hence as categories. The arrow category $Arr(\mathcal{C})$ is equivalently the functor category $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image. By precomposition, this induces functors $d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here

• $d_1$ sends a pair of composable morphisms to their composition;

• $d_2$ sends a pair of composable morphisms to the first morphisms;

• $d_0$ sends a pair of composable morphisms to the second morphisms.

Definition

A weak factorization system, def. , is called a functorial weak factorization system if the factorization of morphisms may be chosen to be a functorial factorization $fact$, def. , i.e. such that $d_2 \circ fact$ lands in $Proj$ and $d_0\circ fact$ in $Inj$.

Remark

Not all weak factorization systems are functorial, def. , although most (including those produced by the small object argument (prop. below), with due care) are.

Proposition

Let $\mathcal{C}$ be a category and let $K\subset Mor(\mathcal{C})$ be a class of morphisms. Write $K Proj$ and $K Inj$, respectively, for the sub-classes of $K$-projective morphisms and of $K$-injective morphisms, def. . Then:

1. Both classes contain the class of isomorphism of $\mathcal{C}$.

2. Both classes are closed under composition in $\mathcal{C}$.

   $K Proj$ is also closed under transfinite composition.

3. Both classes are closed under forming retracts in the arrow category $\mathcal{C}^{\Delta[1]}$ (see remark ).

4. $K Proj$ is closed under forming pushouts of morphisms in $\mathcal{C}$ (“cobase change”).

   $K Inj$ is closed under forming pullback of morphisms in $\mathcal{C}$ (“base change”).

5. $K Proj$ is closed under forming coproducts in $\mathcal{C}^{\Delta[1]}$.

   $K Inj$ is closed under forming products in $\mathcal{C}^{\Delta[1]}$.

Proof

We go through each item in turn.

containing isomorphisms

Given a commuting square

$$\array{ A &\overset{f}{\rightarrow}& X \\ {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}} \\ B &\underset{g}{\longrightarrow}& Y }$$

with the left morphism an isomorphism, then a lift is given by using the inverse of this isomorphism ${}^{{f \circ i^{-1}}}\nearrow$. Hence in particular there is a lift when $p \in K$ and so $i \in K Proj$. The other case is formally dual.

closure under composition

Given a commuting square of the form

$$\array{ A &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y }$$

consider its pasting decomposition as

$$\array{ A &\longrightarrow& X \\ \downarrow &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y } \,.$$

Now the bottom commuting square has a lift, by assumption. This yields another pasting decomposition

$$\array{ A &\longrightarrow& X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y }$$

and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that $p_1\circ p_1$ has the right lifting property against $K$ and is hence in $K Inj$. The case of composing two morphisms in $K Proj$ is formally dual. From this the closure of $K Proj$ under transfinite composition follows since the latter is given by colimits of sequential composition and successive lifts against the underlying sequence as above constitutes a cocone, whence the extension of the lift to the colimit follows by its universal property.

closure under retracts

Let $j$ be the retract of an $i \in K Proj$, i.e. let there be a commuting diagram of the form.

$$\array{ id_A \colon & A &\longrightarrow& C &\longrightarrow& A \\ & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} \\ id_B \colon & B &\longrightarrow& D &\longrightarrow& B } \,.$$

Then for

$$\array{ A &\longrightarrow& X \\ {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& Y }$$

a commuting square, it is equivalent to its pasting composite with that retract diagram

$$\array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,.$$

Here the pasting composite of the two squares on the right has a lift, by assumption:

$$\array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{} && \nearrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,.$$

By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence $j$ has the left lifting property against all $p \in K$ and hence is in $K Proj$. The other case is formally dual.

closure under pushout and pullback

Let $p \in K Inj$ and and let

$$\array{ Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}} \\ Z &\stackrel{f}{\longrightarrow} & Y }$$

be a pullback diagram in $\mathcal{C}$. We need to show that $f^* p$ has the right lifting property with respect to all $i \in K$. So let

$$\array{ A &\longrightarrow& Z \times_f X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}} \\ B &\stackrel{g}{\longrightarrow}& Z }$$

be a commuting square. We need to construct a diagonal lift of that square. To that end, first consider the pasting composite with the pullback square from above to obtain the commuting diagram

$$\array{ A &\longrightarrow& Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,.$$

By the right lifting property of $p$, there is a diagonal lift of the total outer diagram

$$\array{ A &\longrightarrow& X \\ \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{f g}{\longrightarrow}& Y } \,.$$

By the universal property of the pullback this gives rise to the lift $\hat g$ in

$$\array{ && Z \times_f X &\longrightarrow& X \\ &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,.$$

In order for $\hat g$ to qualify as the intended lift of the total diagram, it remains to show that

$$\array{ A &\longrightarrow& Z \times_f X \\ \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow \\ B }$$

commutes. To do so we notice that we obtain two cones with tip $A$:

• one is given by the morphisms

  1. $A \to Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$

  with universal morphism into the pullback being

  • $A \to Z \times_f X$

• the other by

  1. $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$.

  with universal morphism into the pullback being

  • $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X$.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.

The other case is formally dual.

closure under (co-)products

Let $\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S}$ be a set of elements of $K Proj$. Since colimits in the presheaf category $\mathcal{C}^{\Delta[1]}$ are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects $\underset{s \in S}{\coprod} A_s$ induced via its universal property by the set of morphisms $i_s$:

$$\underset{s \in S}{\sqcup} A_s \overset{(i_s)_{s\in S}}{\longrightarrow} \underset{s \in S}{\sqcup} B_s \,.$$

Now let

$$\array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y }$$

be a commuting square. This is in particular a cocone under the coproduct of objects, hence by the universal property of the coproduct, this is equivalent to a set of commuting diagrams

$$\left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in K Proj}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} \,.$$

By assumption, each of these has a lift $\ell_s$. The collection of these lifts

$$\left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in Proj}}\downarrow &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S}$$

is now itself a compatible cocone, and so once more by the universal property of the coproduct, this is equivalent to a lift $(\ell_s)_{s\in S}$ in the original square

$$\array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow &{}^{(\ell_s)_{s\in S}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } \,.$$

This shows that the coproduct of the $i_s$ has the left lifting property against all $f\in K$ and is hence in $K Proj$. The other case is formally dual.

An immediate consequence of prop. is this:

Corollary

Let $\mathcal{C}$ be a category with all small colimits, and let $K\subset Mor(\mathcal{C})$ be a sub-class of its morphisms. Then every $K$-injective morphism, def. , has the right lifting property, def. , against all $K$-relative cell complexes, def. and their retracts, remark .

Remark

By a retract of a morphism $X \stackrel{f}{\longrightarrow} Y$ in some category $\mathcal{C}$ we mean a retract of $f$ as an object in the arrow category $\mathcal{C}^{\Delta[1]}$, hence a morphism $A \stackrel{g}{\longrightarrow} B$ such that in $\mathcal{C}^{\Delta[1]}$ there is a factorization of the identity on $g$ through $f$

$$id_g \;\colon\; g \longrightarrow f \longrightarrow g \,.$$

This means equivalently that in $\mathcal{C}$ there is a commuting diagram of the form

$$\array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,.$$

Lemma

In every category $C$ the class of isomorphisms is preserved under retracts in the sense of remark .

Proof

For

$$\array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,.$$

a retract diagram and $X \overset{f}{\to} Y$ an isomorphism, the inverse to $A \overset{g}{\to} B$ is given by the composite

$$\array{ & & & X & \longrightarrow & A \\ & && \uparrow^{\mathrlap{f^{-1}}} && \\ & B & \longrightarrow& Y&& } \,.$$

More generally:

Proposition

Given a model category in the sense of def. , then its class of weak equivalences is closed under forming retracts (in the arrow category, see remark ).

(Joyal, prop. E.1.3)

Proof

Let

$$\array{ id \colon & A &\longrightarrow& X &\longrightarrow& A \\ & {}^{\mathllap{f}} \downarrow && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{f}} \\ id \colon & B &\longrightarrow& Y &\longrightarrow& B }$$

be a commuting diagram in the given model category, with $w \in W$ a weak equivalence. We need to show that then also $f \in W$.

First consider the case that $f \in Fib$.

In this case, factor $w$ as a cofibration followed by an acyclic fibration. Since $w \in W$ and by two-out-of-three (def. ) this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain a commuting diagram of the following form:

$$\array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{id}} \\ id \colon & A' &\overset{s}{\longrightarrow}& X' &\overset{\phantom{AA}t\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{f}}_{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,,$$

where $s$ is uniquely defined and where $t$ is any lift of the top middle vertical acyclic cofibration against $f$. This now exhibits $f$ as a retract of an acyclic fibration. These are closed under retract by prop. .

Now consider the general case. Factor $f$ as an acyclic cofibration followed by a fibration and form the pushout in the top left square of the following diagram

$$\array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{\in W \cap Cof}}\downarrow &(po)& \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{\in W \cap Cof}} \\ id \colon & A' &\overset{}{\longrightarrow}& X' &\overset{\phantom{AA}\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W }} && \downarrow^{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,,$$

where the other three squares are induced by the universal property of the pushout, as is the identification of the middle horizontal composite as the identity on $A'$. Since acyclic cofibrations are closed under forming pushouts by prop. , the top middle vertical morphism is now an acyclic fibration, and hence by assumption and by two-out-of-three so is the middle bottom vertical morphism.

Thus the previous case now gives that the bottom left vertical morphism is a weak equivalence, and hence the total left vertical composite is.

Lemma

(retract argument)

Consider a composite morphism

$$f \;\colon\; X \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y \,.$$

1. If $f$ has the left lifting property against $p$, then $f$ is a retract of $i$.

2. If $f$ has the right lifting property against $i$, then $f$ is a retract of $p$.

Proof

We discuss the first statement, the second is formally dual.

Write the factorization of $f$ as a commuting square of the form

$$\array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,.$$

By the assumed lifting property of $f$ against $p$ there exists a diagonal filler $g$ making a commuting diagram of the form

$$\array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,.$$

By rearranging this diagram a little, it is equivalent to

$$\array{ & X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,.$$

Completing this to the right, this yields a diagram exhibiting the required retract according to remark :

$$\array{ id_X \colon & X &=& X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow && {}^{\mathllap{f}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,.$$

Small object argument

Given a set $C \subset Mor(\mathcal{C})$ of morphisms in some category $\mathcal{C}$, a natural question is how to factor any given morphism $f\colon X \longrightarrow Y$ through a relative $C$-cell complex, def. , followed by a $C$-injective morphism, def.

$$f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C inj}{\longrightarrow} Y \,.$$

A first approximation to such a factorization turns out to be given simply by forming $\hat X = X_1$ by attaching all possible $C$-cells to $X$. Namely let

$$(C/f) \coloneqq \left\{ \array{ dom(c) &\stackrel{}{\longrightarrow}& X \\ {}^{\mathllap{c\in C}}\downarrow && \downarrow^{\mathrlap{f}} \\ cod(c) &\longrightarrow& Y } \right\}$$

be the set of all ways to find a $C$-cell attachment in $f$, and consider the pushout $\hat X$ of the coproduct of morphisms in $C$ over all these:

$$\array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &(po)& \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& X_1 } \,.$$

This gets already close to producing the intended factorization:

First of all the resulting map $X \to X_1$ is a $C$-relative cell complex, by construction.

Second, by the fact that the coproduct is over all commuting squres to $f$, the morphism $f$ itself makes a commuting diagram

$$\array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow && \downarrow^{\mathrlap{f}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y }$$

and hence the universal property of the colimit means that $f$ is indeed factored through that $C$-cell complex $X_1$; we may suggestively arrange that factorizing diagram like so:

$$\array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} dom(c) && X_1 \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &\nearrow& \downarrow \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y } \,.$$

This shows that, finally, the colimiting co-cone map – the one that now appears diagonally – almost exhibits the desired right lifting of $X_1 \to Y$ against the $c\in C$. The failure of that to hold on the nose is only the fact that a horizontal map in the middle of the above diagram is missing: the diagonal map obtained above lifts not all commuting diagrams of $c\in C$ into $f$, but only those where the top morphism $dom(c) \to X_1$ factors through $X \to X_1$.

The idea of the small object argument now is to fix this only remaining problem by iterating the construction: next factor $X_1 \to Y$ in the same way into

$$X_1 \longrightarrow X_2 \longrightarrow Y$$

and so forth. Since relative $C$-cell complexes are closed under composition, at stage $n$ the resulting $X \longrightarrow X_n$ is still a $C$-cell complex, getting bigger and bigger. But accordingly, the failure of the accompanying $X_n \longrightarrow Y$ to be a $C$-injective morphism becomes smaller and smaller, for it now lifts against all diagrams where $dom(c) \longrightarrow X_n$ factors through $X_{n-1}\longrightarrow X_n$, which intuitively is less and less of a condition as the $X_{n-1}$ grow larger and larger.

The concept of small object is just what makes this intuition precise and finishes the small object argument. For the present purpose we just need the following simple version:

Definition

For $\mathcal{C}$ a category and $C \subset Mor(\mathcal{C})$ a sub-set of its morphisms, say that these have small domains if there is an ordinal $\alpha$ (def. ) such that for every $c\in C$ and for every $C$-relative cell complex given by a transfinite composition (def. )

$$f \;\colon\; X \to X_1 \to X_2 \to \cdots \to X_\beta \to \cdots \longrightarrow \hat X$$

every morphism $dom(c)\longrightarrow \hat X$ factors through a stage $X_\beta \to \hat X$ of order $\beta \lt \alpha$:

$$\array{ && X_\beta \\ & \nearrow & \downarrow \\ dom(c) &\longrightarrow& \hat X } \,.$$

The above discussion proves the following:

Proposition

(small object argument)

Let $\mathcal{C}$ be a locally small category with all small colimits. If a set $C\subset Mor(\mathcal{C})$ of morphisms has all small domains in the sense of def. , then every morphism $f\colon X\longrightarrow$ in $\mathcal{C}$ factors through a $C$-relative cell complex, def. , followed by a $C$-injective morphism, def.

$$f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C inj}{\longrightarrow} Y \,.$$

(Quillen 67, II.3 lemma)

Homotopy

We discuss how the concept of homotopy is abstractly realized in model categories, def. .

Definition

Let $\mathcal{C}$ be a model category, def. , and $X \in \mathcal{C}$ an object.

• A path space object $Path(X)$ for $X$ is a factorization of the diagonal $\Delta_X \;\colon\; X \to X \times X$ as

$$\Delta_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \underoverset{\in Fib}{(p_0,p_1)}{\longrightarrow} X \times X \,.$$

where $X\to Path(X)$ is a weak equivalence and $Path(X) \to X \times X$ is a fibration.

• A cylinder object $Cyl(X)$ for $X$ is a factorization of the codiagonal (or “fold map”) $\nabla_X \;\colon\; X \sqcup X \to X$ as

$$\nabla_X \;\colon\; X \sqcup X \underoverset{\in Cof}{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{\in W}{p}{\longrightarrow} X \,.$$

where $Cyl(X) \to X$ is a weak equivalence. and $X \sqcup X \to Cyl(X)$ is a cofibration.

Remark

For every object $X \in \mathcal{C}$ in a model category, a cylinder object and a path space object according to def. exist: the factorization axioms guarantee that there exists

1. a factorization of the codiagonal as

   $$\nabla_X \;\colon\; X \sqcup X \overset{\in Cof}{\longrightarrow} Cyl(X) \overset{\in W \cap Fib}{\longrightarrow} X$$

2. a factorization of the diagonal as

   $$\Delta_X \;\colon\; X \overset{\in W \cap Cof}{\longrightarrow} Path(X) \overset{\in Fib}{\longrightarrow} X \times X \,.$$

The cylinder and path space objects obtained this way are actually better than required by def. : in addition to $Cyl(X)\to X$ being just a weak equivalence, for these this is actually an acyclic fibration, and dually in addition to $X\to Path(X)$ being a weak equivalence, for these it is actually an acyclic cofibrations.

Some authors call cylinder/path-space objects with this extra property “very good” cylinder/path-space objects, respectively.

One may also consider dropping a condition in def. : what mainly matters is the weak equivalence, hence some authors take cylinder/path-space objects to be defined as in def. but without the condition that $X \sqcup X\to Cyl(X)$ is a cofibration and without the condition that $Path(X) \to X$ is a fibration. Such authors would then refer to the concept in def. as “good” cylinder/path-space objects.

The terminology in def. follows the original (Quillen 67, I.1 def. 4). With the induced concept of left/right homotopy below in def. , this admits a quick derivation of the key facts in the following, as we spell out below.

Lemma

Let $\mathcal{C}$ be a model category. If $X \in \mathcal{C}$ is cofibrant, then for every cylinder object $Cyl(X)$ of $X$, def. , not only is $(i_0,i_1) \colon X \sqcup X \to X$ a cofibration, but each

$$i_0, i_1 \colon X \longrightarrow Cyl(X)$$

is an acyclic cofibration separately.

Dually, if $X \in \mathcal{C}$ is fibrant, then for every path space object $Path(X)$ of $X$, def. , not only is $(p_0,p_1) \colon Path(X)\to X \times X$ a cofibration, but each

$$p_0, p_1 \colon Path(X) \longrightarrow X$$

is an acyclic fibration separately.

Proof

We discuss the case of the path space object. The other case is formally dual.

First, that the component maps are weak equivalences follows generally: by definition they have a right inverse $Path(X) \to X$ and so this follows by two-out-of-three (def. ).

But if $X$ is fibrant, then also the two projection maps out of the product $X \times X \to X$ are fibrations, because they are both pullbacks of the fibration $X \to \ast$

$$\array{ X\times X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ X &\longrightarrow& \ast } \,.$$

hence $p_i \colon Path(X)\to X \times X \to X$ is the composite of two fibrations, and hence itself a fibration, by prop. .

Path space objects are very non-unique as objects up to isomorphism:

Example

If $X \in \mathcal{C}$ is a fibrant object in a model category, def. , and for $Path_1(X)$ and $Path_2(X)$ two path space objects for $X$, def. , then the fiber product $Path_1(X) \times_X Path_2(X)$ is another path space object for $X$: the pullback square

$$\array{ X &\overset{\Delta_X}{\longrightarrow}& X \times X \\ \downarrow && \downarrow \\ Path_1(X) \underset{X}{\times} Path_2(X) &\longrightarrow& Path_1(X)\times Path_2(X) \\ {}^{\mathllap{\in Fib}}\downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ X \times X \times X &\overset{(id,\Delta_X,id)}{\longrightarrow}& X \times X\times X \times X \\ \downarrow^{\mathrlap{(pr_1,pr_3)}}_{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{(p_1, p_4)}} \\ X\times X &=& X \times X }$$

gives that the induced projection is again a fibration. Moreover, using lemma and two-out-of-three (def. ) gives that $X \to Path_1(X) \times_X Path_2(X)$ is a weak equivalence.

For the case of the canonical topological path space objects of def , with $Path_1(X) = Path_2(X) = X^I = X^{[0,1]}$ then this new path space object is $X^{I \vee I} = X^{[0,2]}$, the mapping space out of the standard interval of length 2 instead of length 1.

Definition

(abstract left homotopy and abstract right homotopy

Let $f,g \colon X \longrightarrow Y$ be two parallel morphisms in a model category.

• A left homotopy $\eta \colon f \Rightarrow_L g$ is a morphism $\eta \colon Cyl(X) \longrightarrow Y$ from a cylinder object of $X$, def. , such that it makes this diagram commute:

$$\array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,.$$

• A right homotopy $\eta \colon f \Rightarrow_R g$ is a morphism $\eta \colon X \to Path(Y)$ to some path space object of $X$, def. , such that this diagram commutes:

$$\array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,.$$

Lemma

Let $f,g \colon X \to Y$ be two parallel morphisms in a model category.

1. Let $X$ be cofibrant. If there is a left homotopy $f \Rightarrow_L g$ then there is also a right homotopy $f \Rightarrow_R g$ (def. ) with respect to any chosen path space object.

2. Let $X$ be fibrant. If there is a right homotopy $f \Rightarrow_R g$ then there is also a left homotopy $f \Rightarrow_L g$ with respect to any chosen cylinder object.

In particular if $X$ is cofibrant and $Y$ is fibrant, then by going back and forth it follows that every left homotopy is exhibited by every cylinder object, and every right homotopy is exhibited by every path space object.

Proof

We discuss the first case, the second is formally dual. Let $\eta \colon Cyl(X) \longrightarrow Y$ be the given left homotopy. Lemma implies that we have a lift $h$ in the following commuting diagram

$$\array{ X &\overset{i \circ f}{\longrightarrow}& Path(Y) \\ {}^{\mathllap{i_0}}_{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,,$$

where on the right we have the chosen path space object. Now the composite $\tilde \eta \coloneqq h \circ i_1$ is a right homotopy as required:

$$\array{ && && Path(Y) \\ && &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ X &\overset{i_1}{\longrightarrow}& Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,.$$

Proposition

For $X$ a cofibrant object in a model category and $Y$ a fibrant object, then the relations of left homotopy $f \Rightarrow_L g$ and of right homotopy $f \Rightarrow_R g$ (def. ) on the hom set $Hom(X,Y)$ coincide and are both equivalence relations.

Proof

That both relations coincide under the (co-)fibrancy assumption follows directly from lemma .

The symmetry and reflexivity of the relation is obvious.

That right homotopy (hence also left homotopy) with domain $X$ is a transitive relation follows from using example to compose path space objects.

The homotopy category

We discuss the construction that takes a model category, def. , and then universally forces all its weak equivalences into actual isomorphisms.

Definition

(homotopy category of a model category)

Let $\mathcal{C}$ be a model category, def. . Write $Ho(\mathcal{C})$ for the category whose

• objects are those objects of $\mathcal{C}$ which are both fibrant and cofibrant;

• morphisms are the homotopy classes of morphisms of $\mathcal{C}$, hence the equivalence classes of morphism under the equivalence relation of prop. ;

and whose composition operation is given on representatives by composition in $\mathcal{C}$.

This is, up to equivalence of categories, the homotopy category of the model category $\mathcal{C}$.

Proposition

Def. is well defined, in that composition of morphisms between fibrant-cofibrant objects in $\mathcal{C}$ indeed passes to homotopy classes.

Proof

Fix any morphism $X \overset{f}{\to} Y$ between fibrant-cofibrant objects. Then for precomposition

$$(-) \circ [f] \;\colon\; Hom_{Ho(\mathcal{C})}(Y,Z) \to Hom_{Ho(\mathcal{C}(X,Z))}$$

to be well defined, we need that with $(g\sim h)\;\colon\; Y \to Z$ also $(f g \sim f h)\;\colon\; X \to Z$. But by prop we may take the homotopy $\sim$ to be exhibited by a right homotopy $\eta \colon Y \to Path(Z)$, for which case the statement is evident from this diagram:

$$\array{ && && Z \\ && & {}^{\mathllap{g}}\nearrow & \uparrow^{\mathrlap{p_1}} \\ X &\overset{f}{\longrightarrow} & Y &\overset{\eta}{\longrightarrow}& Path(Z) \\ && & {}_{\mathllap{h}}\searrow & \downarrow_{\mathrlap{p_0}} \\ && && Z } \,.$$

For postcomposition we may choose to exhibit homotopy by left homotopy and argue dually.

We now spell out that def. indeed satisfies the universal property that defines the localization of a category with weak equivalences at its weak equivalences.

Lemma

(Whitehead theorem in model categories)

Let $\mathcal{C}$ be a model category. A weak equivalence between two objects which are both fibrant and cofibrant is a homotopy equivalence (1).

Proof

By the factorization axioms in the model category $\mathcal{C}$ and by two-out-of-three (def. ), every weak equivalence $f\colon X \longrightarrow Y$ factors through an object $Z$ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with $X$ and $Y$ both fibrant and cofibrant, so is $Z$, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.

So let $f \colon X \longrightarrow Y$ be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is formally dual. Then in fact it has a genuine right inverse given by a lift $f^{-1}$ in the diagram

$$\array{ \emptyset &\rightarrow& X \\ {}^{\mathllap{\in cof}}\downarrow &{}^{{f^{-1}}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib \cap W}} \\ X &=& X } \,.$$

To see that $f^{-1}$ is also a left inverse up to left homotopy, let $Cyl(X)$ be any cylinder object on $X$ (def. ), hence a factorization of the codiagonal on $X$ as a cofibration followed by a an acyclic fibration

$$X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{p}{\longrightarrow} X$$

and consider the commuting square

$$\array{ X \sqcup X &\stackrel{(f^{-1}\circ f, id)}{\longrightarrow}& X \\ {}^{\mathllap{\iota_X}}{}_{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{f\circ p}{\longrightarrow}& Y } \,,$$

which commutes due to $f^{-1}$ being a genuine right inverse of $f$. By construction, this commuting square now admits a lift $\eta$, and that constitutes a left homotopy $\eta \colon f^{-1}\circ f \Rightarrow_L id$.

Definition

(fibrant resolution and cofibrant resolution)

Given a model category $\mathcal{C}$, consider a choice for each object $X \in \mathcal{C}$ of

1. a factorization

   $$\emptyset \underoverset{\phantom{A}\in Cof\phantom{A}}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_X}{\longrightarrow} X$$

   of the initial morphism (Def. ), such that when $X$ is already cofibrant then $p_X = id_X$;

2. a factorization

   $$X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} P X \underoverset{\phantom{A} \in Fib \phantom{A}}{q_X}{\longrightarrow} \ast$$

   of the terminal morphism (Def. ), such that when $X$ is already fibrant then $j_X = id_X$.

Write then

$$\gamma_{P,Q} \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$$

for the functor to the homotopy category, def. , which sends an object $X$ to the object $P Q X$ and sends a morphism $f \colon X \longrightarrow Y$ to the homotopy class of the result of first lifting in

$$\array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{i_X}}\downarrow &{}^{Q f}\nearrow& \downarrow^{\mathrlap{p_Y}} \\ Q X &\underset{f\circ p_X}{\longrightarrow}& Y }$$

and then lifting (here: extending) in

$$\array{ Q X &\overset{j_{Q Y} \circ Q f}{\longrightarrow}& P Q Y \\ {}^{\mathllap{j_{Q X}}}\downarrow &{}^{P Q f}\nearrow& \downarrow^{\mathrlap{q_{Q Y}}} \\ P Q X &\longrightarrow& \ast } \,.$$

Lemma

The construction in def. is indeed well defined.

Proof

First of all, the object $P Q X$ is indeed both fibrant and cofibrant (as well as related by a zig-zag of weak equivalences to $X$):

$$\array{ \emptyset \\ {}^{\mathllap{\in Cof}}\downarrow & \searrow^{\mathrlap{\in Cof}} \\ Q X &\underset{\in W \cap Cof}{\longrightarrow}& P Q X &\underset{\in Fib}{\longrightarrow}& \ast \\ {}^{\mathllap{\in W}}\downarrow \\ X } \,.$$

Now to see that the image on morphisms is well defined. First observe that any two choices $(Q f)_{i}$ of the first lift in the definition are left homotopic to each other, exhibited by lifting in

$$\array{ Q X \sqcup Q X &\stackrel{((Q f)_1, (Q f)_2 )}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_{Y}}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(Q X) &\underset{f \circ p_{X} \circ \sigma_{Q X}}{\longrightarrow}& Y } \,.$$

Hence also the composites $j_{Q Y}\circ (Q_f)_i$ are left homotopic to each other, and since their domain is cofibrant, then by lemma they are also right homotopic by a right homotopy $\kappa$. This implies finally, by lifting in

$$\array{ Q X &\overset{\kappa}{\longrightarrow}& Path(P Q Y) \\ {}^{\mathllap{\in W \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib}} \\ P Q X &\underset{(R (Q f)_1, P (Q f)_2)}{\longrightarrow}& P Q Y \times P Q Y }$$

that also $P (Q f)_1$ and $P (Q f)_2$ are right homotopic, hence that indeed $P Q f$ represents a well-defined homotopy class.

Finally to see that the assignment is indeed functorial, observe that the commutativity of the lifting diagrams for $Q f$ and $P Q f$ imply that also the following diagram commutes

$$\array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underset{p_y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& P Q Y } \,.$$

Now from the pasting composite

$$\array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underset{p_Y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& P Q Y \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{Q g}} && \downarrow^{\mathrlap{P Q g}} \\ Z &\underset{p_Z}{\longleftarrow}& Q Z &\underset{j_{Q Z}}{\longrightarrow}& P Q Z }$$

one sees that $(P Q g)\circ (P Q f)$ is a lift of $g \circ f$ and hence the same argument as above gives that it is homotopic to the chosen $P Q(g \circ f)$.

For the following, recall the concept of natural isomorphism between functors: for $F, G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ two functors, then a natural transformation $\eta \colon F \Rightarrow G$ is for each object $c \in Obj(\mathcal{C})$ a morphism $\eta_c \colon F(c) \longrightarrow G(c)$ in $\mathcal{D}$, such that for each morphism $f \colon c_1 \to c_2$ in $\mathcal{C}$ the following is a commuting square:

$$\array{ F(c_1) &\overset{\eta_{c_1}}{\longrightarrow}& G(c_1) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(c_2) &\underset{\eta_{c_2}}{\longrightarrow}& G(c_2) } \,.$$

Such $\eta$ is called a natural isomorphism if its $\eta_c$ are isomorphisms for all objects $c$.

Definition

(localization of a category category with weak equivalences)

For $\mathcal{C}$ a category with weak equivalences, its localization at the weak equivalences is, if it exists,

1. a category denoted $\mathcal{C}[W^{-1}]$

2. a functor

   $$\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]$$

such that

1. $\gamma$ sends weak equivalences to isomorphisms;

2. $\gamma$ is universal with this property, in that:

   for $F \colon \mathcal{C} \longrightarrow D$ any functor out of $\mathcal{C}$ into any category $D$, such that $F$ takes weak equivalences to isomorphisms, it factors through $\gamma$ up to a natural isomorphism $\rho$

   $$\array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) }$$

   and this factorization is unique up to unique isomorphism, in that for $(\tilde F_1, \rho_1)$ and $(\tilde F_2, \rho_2)$ two such factorizations, then there is a unique natural isomorphism $\kappa \colon \tilde F_1 \Rightarrow \tilde F_2$ making the evident diagram of natural isomorphisms commute.

Theorem

(convenient localization of model categories)

For $\mathcal{C}$ a model category, the functor $\gamma_{P,Q}$ in def. (for any choice of $P$ and $Q$) exhibits $Ho(\mathcal{C})$ as indeed being the localization of the underlying category with weak equivalences at its weak equivalences, in the sense of def. :

$$\array{ \mathcal{C} &=& \mathcal{C} \\ {}^{\mathllap{\gamma_{P,Q}}}\downarrow && \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) &\simeq& \mathcal{C}[W^{-1}] } \,.$$

(Quillen 67, I.1 theorem 1)

Proof

First, to see that that $\gamma_{P,Q}$ indeed takes weak equivalences to isomorphisms: By two-out-of-three (def. ) applied to the commuting diagrams shown in the proof of lemma , the morphism $P Q f$ is a weak equivalence if $f$ is:

$$\array{ X &\underoverset{\simeq}{p_X}{\longleftarrow}& Q X &\underoverset{\simeq}{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underoverset{p_y}{\simeq}{\longleftarrow}& Q Y &\underoverset{j_{Q Y}}{\simeq}{\longrightarrow}& P Q Y }$$

With this the “Whitehead theorem for model categories”, lemma , implies that $P Q f$ represents an isomorphism in $Ho(\mathcal{C})$.

Now let $F \colon \mathcal{C}\longrightarrow D$ be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as

$$\array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) }$$

uniquely up to unique natural isomorphism. Now by construction of $P$ and $Q$ in def. , $\gamma_{P,Q}$ is the identity on the full subcategory of fibrant-cofibrant objects. It follows that if $\tilde F$ exists at all, it must satisfy for all $X \stackrel{f}{\to} Y$ with $X$ and $Y$ both fibrant and cofibrant that

$$\tilde F([f]) \simeq F(f) \,,$$

(hence in particular $\tilde F(\gamma_{P,Q}(f)) = F(P Q f)$).

But by def. that already fixes $\tilde F$ on all of $Ho(\mathcal{C})$, up to unique natural isomorphism. Hence it only remains to check that with this definition of $\tilde F$ there exists any natural isomorphism $\rho$ filling the diagram above.

To that end, apply $F$ to the above commuting diagram to obtain

$$\array{ F(X) &\underoverset{iso}{F(p_X)}{\longleftarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(P Q X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{F(Q f)}} && \downarrow^{\mathrlap{F(P Q f)}} \\ F(Y) &\underoverset{F(p_y)}{iso}{\longleftarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(P Q Y) } \,.$$

Here now all horizontal morphisms are isomorphisms, by assumption on $F$. It follows that defining $\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1}$ makes the required natural isomorphism:

$$\array{ \rho_X \colon & F(X) &\underoverset{iso}{F(p_X)^{-1}}{\longrightarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(P Q X) &=& \tilde F(\gamma_{P,Q}(X)) \\ & {}^{\mathllap{F(f)}}\downarrow && && \downarrow^{\mathrlap{F(P Q f)}} && \downarrow^{\tilde F(\gamma_{P,Q}(f))} \\ \rho_Y\colon& F(Y) &\underoverset{F(p_y)^{-1}}{iso}{\longrightarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(P Q Y) &=& \tilde F(\gamma_{P,Q}(X)) } \,.$$

Remark

Due to theorem we may suppress the choices of cofibrant $Q$ and fibrant replacement $P$ in def. and just speak of the localization functor

$$\gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$$

up to natural isomorphism.

In general, the localization $\mathcal{C}[W^{-1}]$ of a category with weak equivalences $(\mathcal{C},W)$ (def. ) may invert more morphisms than just those in $W$. However, if the category admits the structure of a model category $(\mathcal{C},W,Cof,Fib)$, then its localization precisely only inverts the weak equivalences:

Proposition

(localization of model categories inverts precisely the weak equivalences)

Let $\mathcal{C}$ be a model category (def. ) and let $\gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$ be its localization functor (def. , theorem ). Then a morphism $f$ in $\mathcal{C}$ is a weak equivalence precisely if $\gamma(f)$ is an isomorphism in $Ho(\mathcal{C})$.

(e.g. Goerss-Jardine 96, II, prop 1.14)

While the construction of the homotopy category in def. combines the restriction to good (fibrant/cofibrant) objects with the passage to homotopy classes of morphisms, it is often useful to consider intermediate stages:

Definition

Given a model category $\mathcal{C}$, write

$$\array{ && \mathcal{C}_{f c} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} }$$

for the system of full subcategory inclusions of:

1. the category of fibrant objects $\mathcal{C}_f$

2. the category of cofibrant objects $\mathcal{C}_c$,

3. the category of fibrant-cofibrant objects $\mathcal{C}_{fc}$,

all regarded a categories with weak equivalences (def. ), via the weak equivalences inherited from $\mathcal{C}$, which we write $(\mathcal{C}_f, W_f)$, $(\mathcal{C}_c, W_c)$ and $(\mathcal{C}_{f c}, W_{f c})$.

Remark

(categories of fibrant objects and cofibration categories)

Of course the subcategories in def. inherit more structure than just that of categories with weak equivalences from $\mathcal{C}$. $\mathcal{C}_f$ and $\mathcal{C}_c$ each inherit “half” of the factorization axioms. One says that $\mathcal{C}_f$ has the structure of a “fibration category” called a “Brown-category of fibrant objects”, while $\mathcal{C}_c$ has the structure of a “cofibration category”.

We discuss properties of these categories of (co-)fibrant objects below in Homotopy fiber sequences.

The proof of theorem immediately implies the following:

Corollary

For $\mathcal{C}$ a model category, the restriction of the localization functor $\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$ from def. (using remark ) to any of the sub-categories with weak equivalences of def.

$$\array{ && \mathcal{C}_{f c} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} \\ && \downarrow^{\mathrlap{\gamma}} \\ && Ho(\mathcal{C}) }$$

exhibits $Ho(\mathcal{C})$ equivalently as the localization also of these subcategories with weak equivalences, at their weak equivalences. In particular there are equivalences of categories

$$Ho(\mathcal{C}) \simeq \mathcal{C}[W^{-1}] \simeq \mathcal{C}_f[W_f^{-1}] \simeq \mathcal{C}_c[W_c^{-1}] \simeq \mathcal{C}_{f c}[W_{f c}^{-1}] \,.$$

The following says that for computing the hom-sets in the homotopy category, even a mixed variant of the above will do; it is sufficient that the domain is cofibrant and the codomain is fibrant:

Lemma

(hom-sets of homotopy category via mapping cofibrant resolutions into fibrant resolutions)

For $X, Y \in \mathcal{C}$ with $X$ cofibrant and $Y$ fibrant, and for $P, Q$ fibrant/cofibrant replacement functors as in def. , then the morphism

$$Hom_{Ho(\mathcal{C})}(P X,Q Y) = Hom_{\mathcal{C}}(P X, Q Y)/_{\sim} \overset{Hom_{\mathcal{C}}(j_X, p_Y)}{\longrightarrow} Hom_{\mathcal{C}}(X,Y)/_{\sim}$$

(on homotopy classes of morphisms, well defined by prop. ) is a natural bijection.

(Quillen 67, I.1 lemma 7)

Proof

We may factor the morphism in question as the composite

$$Hom_{\mathcal{C}}(P X, Q Y)/_{\sim} \overset{Hom_{\mathcal{C}}(id_{P X}, p_Y)/_\sim }{\longrightarrow} Hom_{\mathcal{C}}(P X, Y)/_{\sim} \overset{Hom_{\mathcal{C}}(j_X, id_Y)/_\sim}{\longrightarrow} Hom_{\mathcal{C}}(X,Y)/_{\sim} \,.$$

This shows that it is sufficient to see that for $X$ cofibrant and $Y$ fibrant, then

$$Hom_{\mathcal{C}}(id_X, p_Y)/_\sim \;\colon\; Hom_{\mathcal{C}}(X, Q Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim$$

is an isomorphism, and dually that

$$Hom_{\mathcal{C}}(j_X, id_Y)/_\sim \;\colon\; Hom_{\mathcal{C}}(P X, Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim$$

is an isomorphism. We discuss this for the former; the second is formally dual:

First, that $Hom_{\mathcal{C}}(id_X, p_Y)$ is surjective is the lifting property in

$$\array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}} \\ X &\overset{f}{\longrightarrow}& Y } \,,$$

which says that any morphism $f \colon X \to Y$ comes from a morphism $\hat f \colon X \to Q Y$ under postcomposition with $Q Y \overset{p_Y}{\to} Y$.

Second, that $Hom_{\mathcal{C}}(id_X, p_Y)$ is injective is the lifting property in

$$\array{ X \sqcup X &\overset{(f,g)}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{\eta}{\longrightarrow}& Y } \,,$$

which says that if two morphisms $f, g \colon X \to Q Y$ become homotopic after postcomposition with $p_Y \colon Q X \to Y$, then they were already homotopic before.

We record the following fact which will be used in part 1.1 (here):

Lemma

Let $\mathcal{C}$ be a model category (def. ). Then every commuting square in its homotopy category $Ho(C)$ (def. ) is, up to isomorphism of squares, in the image of the localization functor $\mathcal{C} \longrightarrow Ho(\mathcal{C})$ of a commuting square in $\mathcal{C}$ (i.e.: not just commuting up to homotopy).

Proof

Let

$$\array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{a}}\downarrow && \downarrow^{\mathrlap{b}} \\ A' &\underset{f'}{\longrightarrow}& B' } \;\;\;\;\; \in Ho(\mathcal{C})$$

be a commuting square in the homotopy category. Writing the same symbols for fibrant-cofibrant objects in $\mathcal{C}$ and for morphisms in $\mathcal{C}$ representing these, then this means that in $\mathcal{C}$ there is a left homotopy of the form

$$\array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{b}} \\ Cyl(A) &\underset{\eta}{\longrightarrow}& B' \\ {}^{\mathllap{i_0}}\uparrow && \uparrow^{\mathrlap{f'}} \\ A &\underset{a}{\longrightarrow}& A' } \,.$$

Consider the factorization of the top square here through the mapping cylinder of $f$

$$\array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{i_1}}\downarrow &(po)& \downarrow^{\mathrlap{\in W}} \\ Cyl(A) &\underset{}{\longrightarrow}& Cyl(f) \\ {}^{\mathllap{i_0}}\uparrow &{}_{\mathllap{\eta}}\searrow& \downarrow^{\mathrlap{}} \\ A && B' \\ & {}_{\mathllap{a}}\searrow & \uparrow_{\mathrlap{f'}} \\ && A' }$$

This exhibits the composite $A \overset{i_0}{\to} Cyl(A) \to Cyl(f)$ as an alternative representative of $f$ in $Ho(\mathcal{C})$, and $Cyl(f) \to B'$ as an alternative representative for $b$, and the commuting square

$$\array{ A &\overset{}{\longrightarrow}& Cyl(f) \\ {}^{\mathllap{a}}\downarrow && \downarrow \\ A' &\underset{f'}{\longrightarrow}& B' }$$

as an alternative representative of the given commuting square in $Ho(\mathcal{C})$.

Derived functors

Definition

(homotopical functor)

For $\mathcal{C}$ and $\mathcal{D}$ two categories with weak equivalences, def. , then a functor $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ is called a homotopical functor if it sends weak equivalences to weak equivalences.

Definition

(derived functor)

Given a homotopical functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ (def. ) between categories with weak equivalences whose homotopy categories $Ho(\mathcal{C})$ and $Ho(\mathcal{D})$ exist (def. ), then its (“total”) derived functor is the functor $Ho(F)$ between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. ):

$$\array{ \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\exists \; Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,.$$

Remark

While many functors of interest between model categories are not homotopical in the sense of def. , many become homotopical after restriction to the full subcategories $\mathcal{C}_f$ of fibrant objects or $\mathcal{C}_c$ of cofibrant objects, def. . By corollary this is just as good for the purpose of homotopy theory.

Therefore one considers the following generalization of def. :

Definition

(left and right derived functors)

Consider a functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ out of a model category $\mathcal{C}$ (def. ) into a category with weak equivalences $\mathcal{D}$ (def. ).

1. If the restriction of $F$ to the full subcategory $\mathcal{C}_f$ of fibrant object becomes a homotopical functor (def. ), then the derived functor of that restriction, according to def. , is called the right derived functor of $F$ and denoted by $\mathbb{R}F$:

   $$\array{ & \mathcal{C}_f &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}} \downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{R} F \colon & \mathcal{C}_f[W^{-1}] &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,,$$

   where we use corollary .

2. If the restriction of $F$ to the full subcategory $\mathcal{C}_c$ of cofibrant object becomes a homotopical functor (def. ), then the derived functor of that restriction, according to def. , is called the left derived functor of $F$ and denoted by $\mathbb{L}F$:

   $$\array{ & \mathcal{C}_c &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{L} F \colon & \mathcal{C}_c[W^{-1}] &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,,$$

   where again we use corollary .

The key fact that makes def. practically relevant is the following:

Proposition

(Ken Brown's lemma)

Let $\mathcal{C}$ be a model category with full subcategories $\mathcal{C}_f, \mathcal{C}_c$ of fibrant objects and of cofibrant objects respectively (def. ). Let $\mathcal{D}$ be a category with weak equivalences.

1. A functor out of the category of fibrant objects

   $$F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D}$$

   is a homotopical functor, def. , already if it sends acyclic fibrations to weak equivalences.

2. A functor out of the category of cofibrant objects

   $$F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D}$$

   is a homotopical functor, def. , already if it sends acyclic cofibrations to weak equivalences.

The following proof refers to the factorization lemma, whose full statement and proof we postpone to further below (lemma ).

Proof

We discuss the case of a functor on a category of fibrant objects $\mathcal{C}_f$, def. . The other case is formally dual.

Let $f \colon X \longrightarrow Y$ be a weak equivalence in $\mathcal{C}_f$. Choose a path space object $Path(X)$ (def. ) and consider the diagram

$$\array{ Path(f) &\underset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{p_1^\ast f}}_{\mathllap{\in W}}\downarrow &(pb)& \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ Path(Y) &\overset{p_1}{\underset{\in W \cap Fib}{\longrightarrow}}& Y \\ {}^{\mathllap{p_0}}_{\mathllap{\in W \cap Fib}}\downarrow \\ Y } \,,$$

where the square is a pullback and $Path(f)$ on the top left is our notation for the universal cone object. (Below we discuss this in more detail, it is the mapping cocone of $f$, def. ).

Here:

1. $p_i$ are both acyclic fibrations, by lemma ;

2. $Path(f) \to X$ is an acyclic fibration because it is the pullback of $p_1$.

3. $p_1^\ast f$ is a weak equivalence, because the factorization lemma states that the composite vertical morphism factors $f$ through a weak equivalence, hence if $f$ is a weak equivalence, then $p_1^\ast f$ is by two-out-of-three (def. ).

Now apply the functor $F$ to this diagram and use the assumption that it sends acyclic fibrations to weak equivalences to obtain

$$\array{ F(Path(f)) &\underset{\in W }{\longrightarrow}& F(X) \\ {}^{\mathllap{F(p_1^\ast f)}}_{\mathllap{}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ F(Path(Y)) &\overset{F(p_1)}{\underset{\in W }{\longrightarrow}}& F(Y) \\ {}^{\mathllap{F(p_0)}}_{\mathllap{\in W}}\downarrow \\ Y } \,.$$

But the factorization lemma , in addition says that the vertical composite $p_0 \circ p_1^\ast f$ is a fibration, hence an acyclic fibration by the above. Therefore also $F(p_0 \circ p_1^\ast f)$ is a weak equivalence. Now the claim that also $F(f)$ is a weak equivalence follows with applying two-out-of-three (def. ) twice.

Corollary

Let $\mathcal{C}, \mathcal{D}$ be model categories and consider $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ a functor. Then:

1. If $F$ preserves cofibrant objects and acyclic cofibrations between these, then its left derived functor (def. ) $\mathbb{L}F$ exists, fitting into a diagram

   $$\array{ \mathcal{C}_{c} &\overset{F}{\longrightarrow}& \mathcal{D}_{c} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\overset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) }$$

2. If $F$ preserves fibrant objects and acyclic fibrants between these, then its right derived functor (def. ) $\mathbb{R}F$ exists, fitting into a diagram

   $$\array{ \mathcal{C}_{f} &\overset{F}{\longrightarrow}& \mathcal{D}_{f} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) } \,.$$

Proposition

(construction of left/right derived functors)

Let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a functor between two model categories (def. ).

1. If $F$ preserves fibrant objects and weak equivalences between fibrant objects, then the total right derived functor $\mathbb{R}F \coloneqq \mathbb{R}(\gamma_{\mathcal{D}}\circ F)$ (def. ) in

   $$\array{ \mathcal{C}_f &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}_f}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) }$$

   is given, up to isomorphism, on any object $X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C})$ by appying $F$ to a fibrant replacement $P X$ of $X$ and then forming a cofibrant replacement $Q(F(P X))$ of the result:

$$\mathbb{R}F(X) \simeq Q(F(P X)) \,.$$

1. If $F$ preserves cofibrant objects and weak equivalences between cofibrant objects, then the total left derived functor $\mathbb{L}F \coloneqq \mathbb{L}(\gamma_{\mathcal{D}}\circ F)$ (def. ) in

   $$\array{ \mathcal{C}_c &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}_c}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) }$$

   is given, up to isomorphism, on any object $X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C})$ by appying $F$ to a cofibrant replacement $Q X$ of $X$ and then forming a fibrant replacement $P(F(Q X))$ of the result:

$$\mathbb{L}F(X) \simeq P(F(Q X)) \,.$$

Proof

We discuss the first case, the second is formally dual. By the proof of theorem we have

$$\begin{aligned} \mathbb{R}F(X) & \simeq \gamma_{\mathcal{D}}(F(\gamma_{\mathcal{C}})) \\ & \simeq \gamma_{\mathcal{D}}F(Q(P(X)) ) \end{aligned} \,.$$

But since $F$ is a homotopical functor on fibrant objects, the cofibrant replacement morphism $F(Q(P(X)))\to F(P(X))$ is a weak equivalence in $\mathcal{D}$, hence becomes an isomorphism under $\gamma_{\mathcal{D}}$. Therefore

$$\mathbb{R}F(X) \simeq \gamma_{\mathcal{D}}(F(P(X))) \,.$$

Now since $F$ is assumed to preserve fibrant objects, $F(P(X))$ is fibrant in $\mathcal{D}$, and hence $\gamma_{\mathcal{D}}$ acts on it (only) by cofibrant replacement.

Quillen adjunctions

In practice it turns out to be useful to arrange for the assumptions in corollary to be satisfied by pairs of adjoint functors (Def. ). Recall that this is a pair of functors $L$ and $R$ going back and forth between two categories

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}$$

such that there is a natural bijection between hom-sets with $L$ on the left and those with $R$ on the right (?):

$$\phi_{d,c} \;\colon\; Hom_{\mathcal{C}}(L(d),c) \underoverset{\simeq}{}{\longrightarrow} Hom_{\mathcal{D}}(d, R(c))$$

for all objects $d\in \mathcal{D}$ and $c \in \mathcal{C}$. This being natural (Def. ) means that $\phi \colon Hom_{\mathcal{D}}(L(-),-) \Rightarrow Hom_{\mathcal{C}}(-, R(-))$ is a natural transformation, hence that for all morphisms $g \colon d_2 \to d_1$ and $f \colon c_1 \to c_2$ the following is a commuting square:

$$\array{ Hom_{\mathcal{C}}(L(d_1), c_1) & \underoverset{\simeq}{\phi_{d_1,c_1}}{\longrightarrow} & Hom_{\mathcal{D}}(d_1, R(c_1)) \\ {}^{\mathllap{L(f) \circ (-)\circ g}}\downarrow && \downarrow^{\mathrlap{g\circ (-)\circ R(g)}} \\ Hom_{\mathcal{C}}(L(d_2), c_2) & \underoverset{\phi_{d_2, c_2}}{\simeq}{\longrightarrow} & Hom_{\mathcal{D}}(d_2, R(c_2)) } \,. /$$

We write $(L \dashv R)$ to indicate such an adjunction and call $L$ the left adjoint and $R$ the right adjoint of the adjoint pair.

The archetypical example of a pair of adjoint functors is that consisting of forming Cartesian products $Y \times (-)$ and forming mapping spaces $(-)^Y$, as in the category of compactly generated topological spaces of def. .

If $f \colon L(d) \to c$ is any morphism, then the image $\phi_{d,c}(f) \colon d \to R(c)$ is called its adjunct, and conversely. The fact that adjuncts are in bijection is also expressed by the notation

$$\frac{ L(c) \overset{f}{\longrightarrow} d }{ c \overset{\tilde f}{\longrightarrow} R(d) } \,.$$

For an object $d\in \mathcal{D}$, the adjunct of the identity on $L d$ is called the adjunction unit $\eta_d \;\colon\; d \longrightarrow R L d$.

For an object $c \in \mathcal{C}$, the adjunct of the identity on $R c$ is called the adjunction counit $\epsilon_c \;\colon\; L R c \longrightarrow c$.

Adjunction units and counits turn out to encode the adjuncts of all other morphisms by the formulas

• $\widetilde{(L d\overset{f}{\to}c)} = (d\overset{\eta}{\to} R L d \overset{R f}{\to} R c)$

• $\widetilde{(d\overset{g}{\to} R c)} = (L d \overset{L g}{\to} L R c \overset{\epsilon}{\to} c)$.

Definition

(Quillen adjunction)

Let $\mathcal{C}, \mathcal{D}$ be model categories. A pair of adjoint functors (Def. ) between them

$$(L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {} \mathcal{D}$$

is called a Quillen adjunction, to be denoted

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

and $L$, $R$ are called left/right Quillen functors, respectively, if the following equivalent conditions are satisfied:

1. $L$ preserves cofibrations and $R$ preserves fibrations;

2. $L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations;

3. $L$ preserves cofibrations and acyclic cofibrations;

4. $R$ preserves fibrations and acyclic fibrations.

Proposition

The conditions in def. are indeed all equivalent.

(Quillen 67, I.4, theorem 3)

Proof

First observe that

• (i) A left adjoint $L$ between model categories preserves acyclic cofibrations precisely if its right adjoint $R$ preserves fibrations.

• (ii) A left adjoint $L$ between model categories preserves cofibrations precisely if its right adjoint $R$ preserves acyclic fibrations.

We discuss statement (i), statement (ii) is formally dual. So let $f\colon A \to B$ be an acyclic cofibration in $\mathcal{D}$ and $g \colon X \to Y$ a fibration in $\mathcal{C}$. Then for every commuting diagram as on the left of the following, its $(L\dashv R)$-adjunct is a commuting diagram as on the right here:

$$\array{ A &\longrightarrow& R(X) \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}} \\ B &\longrightarrow& R(Y) } \;\;\;\;\;\; \,, \;\;\;\;\;\; \array{ L(A) &\longrightarrow& X \\ {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}} \\ L(B) &\longrightarrow& Y } \,.$$

If $L$ preserves acyclic cofibrations, then the diagram on the right has a lift, and so the $(L\dashv R)$-adjunct of that lift is a lift of the left diagram. This shows that $R(g)$ has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if $R$ preserves fibrations, the same argument run from right to left gives that $L$ preserves acyclic fibrations.

Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

The following is the analog of adjunction unit and adjunction counit (Def. ):

Definition

(derived adjunction unit)

Let $\mathcal{C}$ and $\mathcal{D}$ be model categories (Def. ), and let

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}$$

be a Quillen adjunction (Def. ). Then

1. a derived adjunction unit at an object $d \in \mathcal{D}$ is a composition of the form

   $$Q(d) \overset{\eta_{Q(d)}}{\longrightarrow} R(L(Q(d))) \overset{R( j_{L(Q(d))} )}{\longrightarrow} R(P(L(Q(d)))$$

   where

   1. $\eta$ is the ordinary adjunction unit (Def. );

   2. $\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{Q(d)}}{\longrightarrow} Q(d) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{Q(d)}}{\longrightarrow} d$ is a cofibrant resolution in $\mathcal{D}$ (Def. );

   3. $L(Q(d)) \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_{L(Q(d))}}{\longrightarrow} P(L(Q(d))) \underoverset{\in Fib_{\mathcal{C}}}{q_{L(Q(d))}}{\longrightarrow} \ast$ is a fibrant resolution in $\mathcal{C}$ (Def. );

2. a derived adjunction counit at an object $c \in \mathcal{C}$ is a composition of the form

   $$L(Q(R(P(c)))) \overset{ p_{R(P(c))} }{\longrightarrow} L R(P(c)) \overset{\epsilon_{P(c)}}{\longrightarrow} P(c)$$

   where

   1. $\epsilon$ is the ordinary adjunction counit (Def. );

   2. $c \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_c}{\longrightarrow} P c \underoverset{\in Fib_{\mathcal{C}}}{q_c}{\longrightarrow} \ast$ is a fibrant resolution in $\mathcal{C}$ (Def. );

   3. $\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{R(P(c))}}{\longrightarrow} Q(R(P(c))) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{R(P(c))}}{\longrightarrow} R(P(c))$ is a cofibrant resolution in $\mathcal{D}$ (Def. ).

We will see that Quillen adjunctions induce ordinary adjoint pairs of derived functors on homotopy categories (Prop. ). For this we first consider the following technical observation:

Lemma

(right Quillen functors preserve path space objects)

Let $\mathcal{C} \stackrel{\overset{L}{\longleftarrow}}{\underoverset{R}{\bot}{\longrightarrow}} \mathcal{D}$ be a Quillen adjunction, def. .

1. For $X \in \mathcal{C}$ a fibrant object and $Path(X)$ a path space object (def. ), then $R(Path(X))$ is a path space object for $R(X)$.

2. For $X \in \mathcal{C}$ a cofibrant object and $Cyl(X)$ a cylinder object (def. ), then $L(Cyl(X))$ is a cylinder object for $L(X)$.

Proof

Consider the second case, the first is formally dual.

First Observe that $L(Y \sqcup Y) \simeq L Y \sqcup L Y$ because $L$ is left adjoint and hence preserves colimits, hence in particular coproducts.

Hence

$$L(\X \sqcup X \overset{\in Cof}{\to} Cyl(X)) = (L(X) \sqcup L(X) \overset{\in Cof}{\to } L (Cyl(X)))$$

is a cofibration.

Second, with $Y$ cofibrant then also $Y \sqcup Cyl(Y)$ is a cofibrantion, since $Y \to Y \sqcup Y$ is a cofibration (lemma ). Therefore by Ken Brown's lemma (prop. ) $L$ preserves the weak equivalence $Cyl(Y) \overset{\in W}{\longrightarrow} Y$.

Proposition

(derived adjunction)

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a Quillen adjunction, def. , also the corresponding left and right derived functors (Def. , via cor. ) form a pair of adjoint functors

$$Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}R}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot} Ho(\mathcal{D}) \,.$$

Moreover, the adjunction unit and adjunction counit of this derived adjunction are the images of the derived adjunction unit and derived adjunction counit (Def. ) under the localization functors (Theorem ).

(Quillen 67, I.4 theorem 3)

Proof

For the first statement, by def. and lemma it is sufficient to see that for $X, Y \in \mathcal{C}$ with $X$ cofibrant and $Y$ fibrant, then there is a natural bijection

(2)$$Hom_{\mathcal{C}}(L X , Y)/_\sim \simeq Hom_{\mathcal{C}}(X, R Y)/_\sim \,.$$

Since by the adjunction isomorphism for $(L \dashv R)$ such a natural bijection exists before passing to homotopy classes $(-)/_\sim$, it is sufficient to see that this respects homotopy classes. To that end, use from lemma that with $Cyl(Y)$ a cylinder object for $Y$, def. , then $L(Cyl(Y))$ is a cylinder object for $L(Y)$. This implies that left homotopies

$$(f \Rightarrow_L g) \;\colon\; L X \longrightarrow Y$$

given by

$$\eta \;\colon\; Cyl(L X) = L Cyl(X) \longrightarrow Y$$

are in bijection to left homotopies

$$(\tilde f \Rightarrow_L \tilde g) \;\colon\; X \longrightarrow R Y$$

given by

$$\tilde \eta \;\colon\; Cyl(X) \longrightarrow R X \,.$$

This establishes the adjunction. Now regarding the (co-)units: We show this for the adjunction unit, the case of the adjunction counit is formally dual.

First observe that for $d \in \mathcal{D}_c$, then the defining commuting square for the left derived functor from def.

$$\array{ \mathcal{D}_c &\overset{L}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_P}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{P,Q}}} \\ Ho(\mathcal{D}) &\underset{\mathbb{L}L}{\longrightarrow}& Ho(\mathcal{C}) }$$

(using fibrant and fibrant/cofibrant replacement functors $\gamma_P$, $\gamma_{P,Q}$ from def. with their universal property from theorem , corollary ) gives that

$$(\mathbb{L} L ) d \simeq P L P d \simeq P L d \;\;\;\; \in Ho(\mathcal{C}) \,,$$

where the second isomorphism holds because the left Quillen functor $L$ sends the acyclic cofibration $j_d \colon d \to P d$ to a weak equivalence.

The adjunction unit of $(\mathbb{L}L \dashv \mathbb{R}R)$ on $P d \in Ho(\mathcal{C})$ is the image of the identity under

$$Hom_{Ho(\mathcal{C})}((\mathbb{L}L) P d, (\mathbb{L} L) P d) \overset{\simeq}{\to} Hom_{Ho(\mathcal{C})}(P d, (\mathbb{R}R)(\mathbb{L}L) P d) \,.$$

By the above and the proof of prop. , that adjunction isomorphism is equivalently that of $(L \dashv R)$ under the isomorphism

$$Hom_{Ho(\mathcal{C})}(P L d , P L d) \overset{Hom(j_{L d}, id)}{\longrightarrow} Hom_{\mathcal{C}}(L d, P L d)/_\sim$$

of lemma . Hence the derived adjunction unit (Def. ) is the $(L \dashv R)$-adjunct of

$$L d \overset{j_{L d}}{\longrightarrow} P L d \overset{id}{\to} P L d \,,$$

which indeed (by the formula for adjuncts, Prop. ) is the derived adjunction unit

$$X \overset{\eta}{\longrightarrow} R L d \overset{R (j_{L d})}{\longrightarrow} R P L d \,.$$

This suggests to regard passage to homotopy categories and derived functors as itself being a suitable functor from a category of model categories to the category of categories. Due to the role played by the distinction between left Quillen functors and right Quillen functors, this is usefully formulated as a double functor:

Definition

(double category of model categories)

The (very large) double category of model categories $ModCat_{dbl}$ is the double category (Def. ) that has

1. as objects: model categories $\mathcal{C}$ (Def. );

2. as vertical morphisms: left Quillen functors $\mathcal{C} \overset{L}{\longrightarrow} \mathcal{E}$ (Def. );

3. as horizontal morphisms: right Quillen functors $\mathcal{C} \overset{R}{\longrightarrow}\mathcal{D}$ (Def. );

4. as 2-morphisms natural transformations between the composites of underlying functors:

   $$L_2\circ R_1 \overset{\phi}{\Rightarrow} R_2\circ L_1 \phantom{AAAAA} \array{ \mathcal{C} &\overset{\phantom{AA}R_1\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{C} &\underset{\phantom{AA}R_2\phantom{AA}}{\longrightarrow}& \mathcal{D} }$$

and composition is given by ordinary composition of functors, horizontally and vertically, and by whiskering-composition of natural transformations.

(Shulman 07, Example 4.6)

There is hence a forgetful double functor (Remark )

$$F \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)$$

to the double category of squares (Example ) in the 2-category of categories (Example ), which forgets the model category-structure and the Quillen functor-property.

The following records the 2-functoriality of sending Quillen adjunctions to adjoint pairs of derived functors (Prop. ):

Proposition

(homotopy double pseudofunctor on the double category of model categories)

There is a double pseudofunctor (Remark )

$$Ho(-) \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)$$

from the double category of model categories (Def. ) to the double category of squares (Example ) in the 2-category Cat (Example ), which sends

1. a model category $\mathcal{C}$ to its homotopy category of a model category (Def. );

2. a left Quillen functor (Def. ) to its left derived functor (Def. );

3. a right Quillen functor (Def. ) to its right derived functor (Def. );

4. a natural transformation

   $$\array{ \mathcal{C} &\overset{R_1}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{E} &\underset{R_2}{\longrightarrow}& \mathcal{F} }$$

   to the “derived natural transformation”

   $$\array{ Ho(\mathcal{C}) &\overset{\mathbb{R}R_1}{\longrightarrow}& Ho(\mathcal{D}) \\ {}^{\mathllap{\mathbb{L}L_1}}\Big\downarrow &\overset{Ho(\phi)}{\swArrow}& \Big\downarrow{}^{\mathrlap{\mathbb{L}L_2}} \\ Ho(\mathcal{E}) &\underset{\mathbb{R}R_2}{\longrightarrow}& Ho(\mathcal{F}) }$$

   given by the zig-zag

   (3)$$Ho(\phi) \;\colon\; L_2 Q R_1 P \overset{}{\longleftarrow} L_2 Q R_1 Q P \longrightarrow L_2 R_1 Q P \overset{\phi}{\longrightarrow} R_2 L_1 Q P \longrightarrow R_2 P L1 Q P \longleftarrow R_2 R L_1 Q \,,$$

   where the unlabeled morphisms are induced by fibrant resolution $c \to P c$ and cofibrant resolution $Q c \to c$, respectively (Def. ).

(Shulman 07, Theorem 7.6)

Lemma

(recognizing derived natural isomorphisms)

For the derived natural transformation $Ho(\phi)$ in (3) to be invertible in the homotopy category, it is sufficient that for every object $c \in \mathcal{C}$ which is both fibrant and cofibrant the following composite natural transformation

$$R_2 Q L_1 c \overset{ R_2 p_{L_1 c} }{\longrightarrow} R_2 L_1 c \overset{\phi}{\longrightarrow} L_2 R_1 c \overset{ L_2 j_{R_1 c} }{\longrightarrow} L_2 P R_1 c$$

(of $\phi$ with images of fibrant resolution/cofibrant resolution, Def. ) is invertible in the homotopy category, hence that the composite is a weak equivalence (by Prop. ).

(Shulman 07, Remark 7.2)

Example

(derived functor of left-right Quillen functor)

Let $\mathcal{C}$, $\mathcal{D}$ be model categories (Def. ), and let

$$\mathcal{C} \overset{\phantom{A}F\phantom{A}}{\longrightarrow} \mathcal{C}$$

be a functor that is both a left Quillen functor as well as a right Quillen functor (Def. ). This means equivalently that there is a 2-morphism in the double category of model categories (Def. ) of the form

(4)$$\array{ \mathcal{C} &\overset{\phantom{AA}F\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{F}}\Big\downarrow &{}^{id}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{D} &\underset{\phantom{A}id\phantom{A}}{\longrightarrow}& \mathcal{D} }$$

It follows that the left derived functor $\mathbb{L}F$ and right derived functor $\mathbb{R}F$ of $F$ (Def. ) are naturally isomorphic:

$$Ho(\mathcal{C}) \overset{ \mathbb{L}F \simeq \mathbb{R}F }{\longrightarrow} Ho(\mathcal{D}) \,.$$

(Shulman 07, corollary 7.8)

Proof

To see the natural isomorphism $\mathbb{L}F \simeq \mathbb{R}F$: By Prop. this is implied once the derived natural transformation $Ho(id)$ of (4) is a natural isomorphism. By Prop. this is the case, in the present situation, if the composition of

$$Q F c \overset{ p_{F c} }{\longrightarrow} F c \overset{ j_{F c} }{\longrightarrow} P F c$$

is a weak equivalence. But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution (Def. ).

The following is the analog of co-reflective subcategories (Def. ) for model categories:

Definition

(Quillen reflection)

Let $\mathcal{C}$ and $\mathcal{D}$ be model categories (Def. ), and let

$$\mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}$$

be a Quillen adjunction between them (Def. ). Then this may be called

1. a Quillen reflection if the derived adjunction counit (Def. ) is componentwise a weak equivalence;

2. a Quillen co-reflection if the derived adjunction unit (Def. ) is componentwise a weak equivalence.

The main class of examples of Quillen reflections are left Bousfield localizations, discussed as Prop. below.

Proposition

(characterization of Quillen reflections)

Let

$$\mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}$$

be a Quillen adjunction (Def. ) and write

$$Ho(\mathcal{C}) \underoverset {\underset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot_{Qu}} Ho(\mathcal{D})$$

for the induced adjoint pair of derived functors on the homotopy categories, from Prop. .

Then

1. $(L \underset{Qu}{\dashv} R)$ is a Quillen reflection (Def. ) precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a reflective subcategory-inclusion (Def. );

2. $(L \underset{Qu}{\dashv} R)$ is a Quillen co-reflection] (Def. ) precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a co-reflective subcategory-inclusion (Def. );

Proof

By Prop. the components of the adjunction unit/counit of $(\mathbb{L}L \dashv \mathbb{R}R)$ are precisely the images under localization of the derived adjunction unit/counit of $(L \underset{Qu}{\dashv} R)$. Moreover, by Prop. the localization functor of a model category inverts precisely the weak equivalences. Hence the adjunction (co-)unit of $(\mathbb{L}L \dashv \mathbb{R}R)$ is an isomorphism if and only if the derived (co-)unit of $(L \underset{Qu}{\dashv} R)$ is a weak equivalence, respectively.

With this the statement reduces to the characterization of (co-)reflections via invertible units/counits, respectively, from Prop. .

The following is the analog of adjoint equivalence of categories (Def. ) for model categories:

Definition

(Quillen equivalence)

For $\mathcal{C}, \mathcal{D}$ two model categories (Def. ), a Quillen adjunction (def. )

$$\mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

is called a Quillen equivalence, to be denoted

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {{}_{\phantom{Qu}}\simeq_{Qu}} \mathcal{D} \,,$$

if the following equivalent conditions hold:

1. The right derived functor of $R$ (via prop. , corollary ) is an equivalence of categories

   $$\mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D}) \,.$$

2. The left derived functor of $L$ (via prop. , corollary ) is an equivalence of categories

   $$\mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C}) \,.$$

3. For every cofibrant object $d\in \mathcal{D}$, the derived adjunction unit (Def. )

   $$d \overset{\eta_d}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d)))$$

   is a weak equivalence;

   and for every fibrant object $c \in \mathcal{C}$, the derived adjunction counit (Def. )

   $$L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c$$

   is a weak equivalence.

4. For every cofibrant object $d \in \mathcal{D}$ and every fibrant object $c \in \mathcal{C}$, a morphism $d \longrightarrow R(c)$ is a weak equivalence precisely if its adjunct morphism $L(c) \to d$ is:

   $$\frac{ d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c) }{ L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c } \,.$$

Poposition

The conditions in def. are indeed all equivalent.

(Quillen 67, I.4, theorem 3)

Proof

That $1) \Leftrightarrow 2)$ follows from prop. (if in an adjoint pair one is an equivalence, then so is the other).

To see the equivalence $1),2) \Leftrightarrow 3)$, notice (prop.) that a pair of adjoint functors is an equivalence of categories precisely if both the adjunction unit and the adjunction counit are natural isomorphisms. Hence it is sufficient to see that the derived adjunction unit/derived adjunction counit (Def. ) indeed represent the adjunction (co-)unit of $(\mathbb{L}L \dashv \mathbb{R}R)$ in the homotopy category. But this is the statement of Prop. .

To see that $4) \Rightarrow 3)$:

Consider the weak equivalence $L X \overset{j_{L X}}{\longrightarrow} P L X$. Its $(L \dashv R)$-adjunct is

$$X \overset{\eta}{\longrightarrow} R L X \overset{R j_{L X}}{\longrightarrow} R P L X$$

by assumption 4) this is again a weak equivalence, which is the requirement for the derived adjunction unit in 3). Dually for derived adjunction counit.

To see $3) \Rightarrow 4)$:

Consider any $f \colon L d \to c$ a weak equivalence for cofibrant $d$, firbant $c$. Its adjunct $\tilde f$ sits in a commuting diagram

$$\array{ \tilde f \colon & d &\overset{\eta}{\longrightarrow}& R L d &\overset{R f}{\longrightarrow}& R c \\ & {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{R j_{L d}}} && \downarrow^{\mathrlap{R j_c}} \\ & d &\underset{\in W}{\longrightarrow}& R P L d &\overset{R P f}{\longrightarrow}& R P c } \,,$$

where $P f$ is any lift constructed as in def. .

This exhibits the bottom left morphism as the derived adjunction unit (Def. ), hence a weak equivalence by assumption. But since $f$ was a weak equivalence, so is $P f$ (by two-out-of-three). Thereby also $R P f$ and $R j_Y$, are weak equivalences by Ken Brown's lemma and the assumed fibrancy of $c$. Therefore by two-out-of-three (def. ) also the adjunct $\tilde f$ is a weak equivalence.

Example

(trivial Quillen equivalence)

Let $\mathcal{C}$ be a model category (Def. ). Then the identity functor on $\mathcal{C}$ constitutes a Quillen equivalence (Def. ) from $\mathcal{C}$ to itself:

$$\mathcal{C} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}$$

Proof

From prop. it is clear that in this case the derived functors $\mathbb{L}id$ and $\mathbb{R}id$ both are themselves the identity functor on the homotopy category of a model category, hence in particular are an equivalence of categories.

In certain situations the conditions on a Quillen equivalence simplify. For instance:

Proposition

(recognition of Quillen equivalences)

If in a Quillen adjunction $\array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}$ (def. ) the right adjoint $R$ “creates weak equivalences” (in that a morphism $f$ in $\mathcal{C}$ is a weak equivalence precisly if $U(f)$ is) then $(L \dashv R)$ is a Quillen equivalence (def. ) precisely already if for all cofibrant objects $d \in \mathcal{D}$ the plain adjunction unit

$$d \overset{\eta}{\longrightarrow} R (L (d))$$

is a weak equivalence.

Proof

By prop. , generally, $(L \dashv R)$ is a Quillen equivalence precisely if

1. for every cofibrant object $d\in \mathcal{D}$, the derived adjunction unit (Def. )

   $$d \overset{\eta}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d)))$$

   is a weak equivalence;

2. for every fibrant object $c \in \mathcal{C}$, the derived adjunction counit (Def. )

   $$L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c$$

   is a weak equivalence.

Consider the first condition: Since $R$ preserves the weak equivalence $j_{L(d)}$, then by two-out-of-three (def. ) the composite in the first item is a weak equivalence precisely if $\eta$ is.

Hence it is now sufficient to show that in this case the second condition above is automatic.

Since $R$ also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image

$$R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c)$$

under $R$ is.

Moreover, assuming, by the above, that $\eta_{Q(R(c))}$ on the cofibrant object $Q(R(c))$ is a weak equivalence, then by two-out-of-three this composite is a weak equivalence precisely if the further composite with $\eta$ is

$$Q(R(c)) \overset{\eta_{Q(R(c))}}{\longrightarrow} R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) \,.$$

By the formula for adjuncts, this composite is the $(L\dashv R)$-adjunct of the original composite, which is just $p_{R(c)}$

$$\frac{ L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c }{ Q(R(C)) \overset{p_{R(c)}}{\longrightarrow} R(c) } \,.$$

But $p_{R(c)}$ is a weak equivalence by definition of cofibrant replacement.

The following is the analog of adjoint triples, adjoint quadruples (Remark ), etc. for model categories:

Definition

(Quillen adjoint triple)

Let $\mathcal{C}_1, \mathcal{C}_2, \mathcal{D}$ be model categories (Def. ), where $\mathcal{C}_1$ and $\mathcal{C}_2$ share the same underlying category $\mathcal{C}$, and such that the identity functor on $\mathcal{C}$ constitutes a Quillen equivalence (Def. ):

$$\mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{C}_1$$

Then

1. a Quillen adjoint triple of the form

   $$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D}$$

   is diagrams in the double category of model categories (Def. ) of the form

   $$\array{ && \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 \\ && {}^{\mathllap{ L }}\Big\downarrow &{}^{\mathllap{\eta}}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\overset{ \phantom{A}R\phantom{A} }{\longrightarrow}& \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{ id }}\Big\downarrow & {}^{\mathllap{\epsilon}}\swArrow & {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 }$$

   such that $\eta$ is the unit of an adjunction and $\epsilon$ the counit of an adjunction, thus exhibiting Quillen adjunctions

   $$\array{ \mathcal{C}_1 \underoverset {\underset{C}{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }$$

   and such that the derived natural transformation $Ho(id)$ of the bottom right square (3) is invertible (a natural isomorphism);

2. a Quillen adjoint triple of the form

   $$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longrightarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longleftarrow} \\ } \mathcal{D}$$

   is diagram in the double category of model categories (Def. ) of the form

   $$\array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 }$$

   such that $\eta$ is the unit of an adjunction and $\epsilon$ the counit of an adjunction, thus exhibiting Quillen adjunctions

   $$\array{ \mathcal{C}_2 \underoverset {\underset{C}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_1 \underoverset {\underset{R}{\longleftarrow}} {\overset{C}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }$$

   and such that the derived natural transformation $Ho(id)$ of the top left square square (here) is invertible (a natural isomorphism).

If a Quillen adjoint triple of the first kind overlaps with one of the second kind

$$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L_1 \phantom{= A_a}}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C_1 = L_2}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{R_1 = C_2}{\longrightarrow} \\ \overset{\phantom{A_a = } R_2}{\longleftarrow} \\ } \mathcal{D}$$

we speak of a Quillen adjoint quadruple, and so forth.

Proposition

(Quillen adjoint triple induces adjoint triple of derived functors on homotopy categories)

Given a Quillen adjoint triple (Def. ), the induced derived functors (Def. ) on the homotopy categories form an ordinary adjoint triple (Remark ):

$$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})$$

$\,$

$$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})$$

Proof

This follows immediately from the fact that passing to homotopy categories of model categories is a double pseudofunctor from the double category of model categories to the double category of squares in Cat (Prop. ).

$\,$

Mapping cones

In the context of homotopy theory, a pullback diagram, such as in the definition of the fiber in example

$$\array{ fib(f) &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y }$$

ought to commute only up to a (left/right) homotopy (def. ) between the outer composite morphisms. Moreover, it should satisfy its universal property up to such homotopies.

Instead of going through the full theory of what this means, we observe that this is plausibly modeled by the following construction, and then we check (below) that this indeed has the relevant abstract homotopy theoretic properties.

Definition

Let $\mathcal{C}$ be a model category, def. with $\mathcal{C}^{\ast/}$ its model structure on pointed objects, prop. . For $f \colon X \longrightarrow Y$ a morphism between cofibrant objects (hence a morphism in $(\mathcal{C}^{\ast/})_c\hookrightarrow \mathcal{C}^{\ast/}$, def. ), its reduced mapping cone is the object

$$Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y$$

in the colimiting diagram

$$\array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) \\ \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow \\ {*} &\longrightarrow& &\longrightarrow& Cone(f) } \,,$$

where $Cyl(X)$ is a cylinder object for $X$, def. .

Dually, for $f \colon X \longrightarrow Y$ a morphism between fibrant objects (hence a morphism in $(\mathcal{C}^{\ast})_f\hookrightarrow \mathcal{C}^{\ast/}$, def. ), its mapping cocone is the object

$$Path_\ast(f) \coloneqq \ast \underset{Y}{\times} Path(Y)\underset{Y}{\times} Y$$

in the following limit diagram

$$\array{ Path_\ast(f) &\longrightarrow& &\longrightarrow& X \\ \downarrow &\searrow^{\mathrlap{\eta}}& && \downarrow^{\mathrlap{f}} \\ && Path(Y) &\underset{p_1}{\longrightarrow}& Y \\ \downarrow && \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& Y } \,,$$

where $Path(Y)$ is a path space object for $Y$, def. .

Remark

When we write homotopies (def. ) as double arrows between morphisms, then the limit diagram in def. looks just like the square in the definition of fibers in example , except that it is filled by the right homotopy given by the component map denoted $\eta$:

$$\array{ Path_\ast(f) &\longrightarrow& X \\ \downarrow &\swArrow_{\eta}& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } \,.$$

Dually, the colimiting diagram for the mapping cone turns to look just like the square for the cofiber, except that it is filled with a left homotopy

$$\array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &\swArrow_{\eta}& \downarrow \\ \ast &\longrightarrow& Cone(f) }$$

Proposition

The colimit appearing in the definition of the reduced mapping cone in def. is equivalent to three consecutive pushouts:

$$\array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f) } \,.$$

The two intermediate objects appearing here are called

• the plain reduced cone $Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)$;

• the reduced mapping cylinder $Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y$.

Dually, the limit appearing in the definition of the mapping cocone in def. is equivalent to three consecutive pullbacks:

$$\array{ Path_\ast(f) &\longrightarrow& Path(f) &\longrightarrow& X \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Path_\ast(Y) &\longrightarrow& Path(Y) &\underset{p_1}{\longrightarrow}& Y \\ \downarrow &(pb)& \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& Y } \,.$$

The two intermediate objects appearing here are called

• the based path space object $Path_\ast(Y) \coloneqq \ast \underset{Y}{\prod} Path(Y)$;

• the mapping path space or mapping co-cylinder $Path(f) \coloneqq X \underset{Y}{\times} Path(X)$.

Definition

Let $X \in \mathcal{C}^{\ast/}$ be any pointed object.

1. The mapping cone, def. , of $X \to \ast$ is called the reduced suspension of $X$, denoted

   $$\Sigma X = Cone(X\to\ast)\,.$$

   Via prop. this is equivalently the coproduct of two copies of the cone on $X$ over their base:

   $$\array{ && X &\stackrel{}{\longrightarrow}& \ast \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X } \,.$$

   This is also equivalently the cofiber, example of $(i_0,i_1)$, hence (example ) of the wedge sum inclusion:

   $$X \vee X = X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \overset{cofib(i_0,i_1)}{\longrightarrow} \Sigma X \,.$$

2. The mapping cocone, def. , of $\ast \to X$ is called the loop space object of $X$, denoted

   $$\Omega X = Path_\ast(\ast \to X) \,.$$

   Via prop. this is equivalently

   $$\array{ \Omega X &\longrightarrow& Path_\ast(X) &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{} \\ Path_\ast(X) &\longrightarrow& Path(X) &\underset{p_1}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& X } \,.$$

   This is also equivalently the fiber, example of $(p_0,p_1)$:

   $$\Omega X \overset{fib(p_0,p_1)}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X \,.$$

Proposition

In pointed topological spaces $Top^{\ast/}$,

• the reduced suspension objects (def. ) induced from the standard reduced cylinder $(-)\wedge (I_+)$ of example are isomorphic to the smash product (def. ) with the 1-sphere, for later purposes we choose to smash on the left and write

  $$cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,,$$

Dually:

• the loop space objects (def. ) induced from the standard pointed path space object $Maps(I_+,-)_\ast$ are isomorphic to the pointed mapping space (example ) with the 1-sphere

  $$fib(Maps(I_+,X)_\ast \to X \times X) \simeq Maps(S^1, X)_\ast \,.$$

Proof

By immediate inspection: For instance the fiber of $Maps(I_+,X)_\ast \longrightarrow X\times X$ is clearly the subspace of the unpointed mapping space $X^I$ on elements that take the endpoints of $I$ to the basepoint of $X$.

Example

For $\mathcal{C} =$ Top with $Cyl(X) = X\times I$ the standard cyclinder object, def. , then by example , the mapping cone, def. , of a continuous function $f \colon X \longrightarrow Y$ is obtained by

1. forming the cylinder over $X$;

2. attaching to one end of that cylinder the space $Y$ as specified by the map $f$.

3. shrinking the other end of the cylinder to the point.

Accordingly the suspension of a topological space is the result of shrinking both ends of the cylinder on the object two the point. This is homeomoprhic to attaching two copies of the cone on the space at the base of the cone.

(graphics taken from Muro 2010)

Below in example we find the homotopy-theoretic interpretation of this standard topological mapping cone as a model for the homotopy cofiber.

Remark

The formula for the mapping cone in prop. (as opposed to that of the mapping co-cone) does not require the presence of the basepoint: for $f \colon X \longrightarrow Y$ a morphism in $\mathcal{C}$ (as opposed to in $\mathcal{C}^{\ast/}$) we may still define

$$Cone'(f) \coloneqq Y \underset{X}{\sqcup} Cone'(X) \,,$$

where the prime denotes the unreduced cone, formed from a cylinder object in $\mathcal{C}$.

Proposition

For $f \colon X \longrightarrow Y$ a morphism in Top, then its unreduced mapping cone, remark , with respect to the standard cylinder object $X \times I$ def. , is isomorphic to the reduced mapping cone, def. , of the morphism $f_+ \colon X_+ \to Y_+$ (with a basepoint adjoined, def. ) with respect to the standard reduced cylinder (example ):

$$Cone'(f) \simeq Cone(f_+) \,.$$

Proof

By prop. and example , $Cone(f_+)$ is given by the colimit in $Top$ over the following diagram:

$$\array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone(f_+) } \,.$$

We may factor the vertical maps to give

$$\array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+ \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone'(f) } \,.$$

This way the top part of the diagram (using the pasting law to compute the colimit in two stages) is manifestly a cocone under the result of applying $(-)_+$ to the diagram for the unreduced cone. Since $(-)_+$ is itself given by a colimit, it preserves colimits, and hence gives the partial colimit $Cone'(f)_+$ as shown. The remaining pushout then contracts the remaining copy of the point away.

Example makes it clear that every cycle $S^n \to Y$ in $Y$ that happens to be in the image of $X$ can be continuously translated in the cylinder-direction, keeping it constant in $Y$, to the other end of the cylinder, where it shrinks away to the point. This means that every homotopy group of $Y$, def. , in the image of $f$ vanishes in the mapping cone. Hence in the mapping cone the image of $X$ under $f$ in $Y$ is removed up to homotopy. This makes it intuitively clear how $Cone(f)$ is a homotopy-version of the cokernel of $f$. We now discuss this formally.

Lemma

(factorization lemma)

Let $\mathcal{C}_c$ be a category of cofibrant objects, def. . Then for every morphism $f \colon X \longrightarrow Y$ the mapping cylinder-construction in def. provides a cofibration resolution of $f$, in that

1. the composite morphism $X \overset{i_0}{\longrightarrow} Cyl(X) \overset{(i_1)_\ast f}{\longrightarrow} Cyl(f)$ is a cofibration;

2. $f$ factors through this morphism by a weak equivalence left inverse to an acyclic cofibration

   $$f \;\colon\; X \underoverset{\in Cof}{(i_1)_\ast f\circ i_0}{\longrightarrow} Cyl(f) \underset{\in W}{\longrightarrow} Y \,,$$

Dually:

Let $\mathcal{C}_f$ be a category of fibrant objects, def. . Then for every morphism $f \colon X \longrightarrow Y$ the mapping cocylinder-construction in def. provides a fibration resolution of $f$, in that

1. the composite morphism $Path(f) \overset{p_1^\ast f}{\longrightarrow} Path(Y) \overset{p_0}{\longrightarrow} Y$ is a fibration;

2. $f$ factors through this morphism by a weak equivalence right inverse to an acyclic fibration:

   $$f \;\colon\; X \underset{\in W}{\longrightarrow} Path(f) \underoverset{\in Fib}{p_0 \circ p_1^\ast f}{\longrightarrow} Y \,,$$

Proof

We discuss the second case. The first case is formally dual.

So consider the mapping cocylinder-construction from prop.

$$\array{ Path(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{p_1^\ast f}}\downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Path(Y) &\underoverset{\in W \cap Fib}{p_1}{\longrightarrow}& Y \\ {}^{\mathllap{\in W \cap Fib}}\downarrow^{\mathrlap{p_0}} \\ Y } \,.$$

To see that the vertical composite is indeed a fibration, notice that, by the pasting law, the above pullback diagram may be decomposed as a pasting of two pullback diagram as follows

$$\array{ Path(f) &\underoverset{\in Fib}{(f,id)^\ast(p_1,p_0)}{\longrightarrow}& X \times Y &\stackrel{pr_1}{\to}& X \\ \downarrow && \downarrow^{\mathrlap{(f, Id)}} && \downarrow^\mathrlap{f} \\ Path(Y) &\overset{(p_1,p_0) \in Fib }{\longrightarrow}& Y \times Y &\stackrel{pr_1}{\longrightarrow}& Y \\ {}^{\mathllap{p_0}}\downarrow & \swarrow_{\mathrlap{pr_2 \atop {\in Fib}}} \\ Y } \,.$$

Both squares are pullback squares. Since pullbacks of fibrations are fibrations by prop. , the morphism $Path(f) \to X \times Y$ is a fibration. Similarly, since $X$ is fibrant, also the projection map $X \times Y \to Y$ is a fibration (being the pullback of $X \to \ast$ along $Y \to \ast$).

Since the vertical composite is thereby exhibited as the composite of two fibrations

$$Path(f) \overset{(f,id)^\ast(p_1,p_0)}{\longrightarrow} X \times Y \stackrel{pr_2 \circ (f ,Id) = pr_2}{\longrightarrow} Y \,,$$

it is itself a fibration.

Then to see that there is a weak equivalence as claimed:

The universal property of the pullback $Path(f)$ induces a right inverse of $Path(f) \to X$ fitting into this diagram

$$\array{ id_X \colon & X &\underoverset{\in W}{\exists}{\longrightarrow} & Path(f) & \overset{\in W \cap Fib}{\longrightarrow}& X \\ & {}^{\mathrlap{f}}\downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ id_Y\colon& Y &\underoverset{\in W}{i}{\longrightarrow}& Path(Y) &\stackrel{p_1}{\to}& Y \\ & & {}_{\mathllap{Id}}\searrow& \downarrow^{\mathrlap{p_0}} \\ & && Y } \,,$$

which is a weak equivalence, as indicated, by two-out-of-three (def. ).

This establishes the claim.

Categories of fibrant objects

Below we discuss the homotopy-theoretic properties of the mapping cone- and mapping cocone-constructions from above. Before we do so, we here establish a collection of general facts that hold in categories of fibrant objects and dually in categories of cofibrant objects, def. .

Literature (Brown 73, section 4).

Lemma

Let $f\colon X \longrightarrow Y$ be a morphism in a category of fibrant objects, def. . Then given any choice of path space objects $Path(X)$ and $Path(Y)$, def. , there is a replacement of $Path(X)$ by a path space object $\widetilde{Path(X)}$ along an acylic fibration, such that $\widetilde{Path(X)}$ has a morphism $\phi$ to $Path(Y)$ which is compatible with the structure maps, in that the following diagram commutes

$$\array{ && X &\overset{f}{\longrightarrow}& Y \\ &\swarrow& \downarrow && \downarrow \\ Path(X) &\underset{\in W \cap Fib}{\longleftarrow}& \widetilde{Path(X)} &\overset{\phi}{\longrightarrow}& Path(Y) \\ &{}_{\mathllap{(p^X_0,p^X_1)}}\searrow& \downarrow^{\mathrlap{(p^Y_0,p^Y_1)}} && \downarrow^{\mathrlap{(\tilde p^X_0,\tilde p^X_1)}} \\ && X \times X &\overset{(f,f)}{\longrightarrow}& Y \times Y } \,.$$

(Brown 73, section 2, lemma 2)

Proof

Consider the commuting square

$$\array{ X &\overset{f}{\longrightarrow}& Y &\longrightarrow& Path(Y) \\ \downarrow && && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}} \\ Path(X) &\overset{(p^X_0,p^X_1)}{\longrightarrow}& X \times X &\overset{(f,f)}{\longrightarrow}& Y \times Y } \,.$$

Then consider its factorization through the pullback of the right morphism along the bottom morphism,

$$\array{ X &\longrightarrow& (f \circ p_0^X, f\circ p_1^X)^\ast Path(Y) &\longrightarrow& Path(Y) \\ &{}_{\mathllap{\in W}}\searrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}}_{\mathrlap{\in Fib}} \\ && Path(X) &\overset{(f \circ p_0^X, f\circ p_1^X)}{\longrightarrow}& Y \times Y } \,.$$

Finally use the factorization lemma to factor the morphism $X \to (f \circ p_0^X, f\circ p_1^X)^\ast Path(Y)$ through a weak equivalence followed by a fibration, the object this factors through serves as the desired path space resolution

$$\array{ X &\overset{\in W}{\longrightarrow}& \widetilde{Path(X)} &\longrightarrow& Path(Y) \\ &{}_{\mathllap{\in W}}\searrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}} \\ && Path(X) &\overset{(f \circ p_0^X, f\circ p_1^X)}{\longrightarrow}& Y \times Y } \,.$$

Lemma

In a category of fibrant objects $\mathcal{C}_f$, def. , let

$$\array{ A_1 &&\stackrel{f}{\longrightarrow}&& A_2 \\ & {}_{\in Fib}\searrow && \swarrow_{\in Fib} \\ && B }$$

be a morphism over some object $B$ in $\mathcal{C}_f$ and let $u \colon B' \to B$ be any morphism in $\mathcal{C}_f$. Let

$$\array{ u^*A_1 &&\stackrel{u^* f}{\longrightarrow}&& u^* A_2 \\ & {}_{\in Fib}\searrow && \swarrow_{\in Fib} \\ && B' }$$

be the corresponding morphism pulled back along $u$.

Then

• if $f$ is a fibration then also $u^* f$ is a fibration;

• if $f$ is a weak equivalence then also $u^* f$ is a weak equivalence.

(Brown 73, section 4, lemma 1)

Proof

For $f \in Fib$ the statement follows from the pasting law which says that if in

$$\array{ B' \times_B A_1 &\longrightarrow& A_1 \\ \;\;\downarrow^{\mathrlap{u^* f \in Fib}} && \;\;\downarrow^{\mathrlap{f \in Fib}} \\ B' \times_B A_2 &\longrightarrow& A_2 \\ \;\downarrow^{\mathrlap{\in Fib}} && \;\downarrow^{\mathrlap{\in Fib}} \\ B' &\stackrel{u}{\longrightarrow}& B }$$

the bottom and the total square are pullback squares, then so is the top square. The same reasoning applies for $f \in W \cap Fib$.

Now to see the case that $f\in W$:

Consider the full subcategory $(\mathcal{C}_{/B})_f$ of the slice category $\mathcal{C}_{/B}$ (def. ) on its fibrant objects, i.e. the full subcategory of the slice category on the fibrations

$$\array{ X \\ \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib}} \\ B }$$

into $B$. By factorizing for every such fibration the diagonal morphisms into the fiber product $X \underset{B}{\times} X$ through a weak equivalence followed by a fibration, we obtain path space objects $Path_B(X)$ relative to $B$:

$$\array{ (\Delta_X)/B \;\colon & X &\overset{\in W}{\longrightarrow}& Path_B(X) &\overset{\in Fib}{\longrightarrow}& X \underset{B}{\times} X \\ & & {}_{\mathllap{\in Fib}}\searrow & \downarrow & \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,.$$

With these, the factorization lemma (lemma ) applies in $(\mathcal{C}_{/B})_f$.

(Notice that for this we do need the restriction of $\mathcal{C}_{/B}$ to the fibrations, because this ensures that the projections $p_i \colon X_1 \times_B X_2 \to X_i$ are still fibrations, which is used in the proof of the factorization lemma (here).)

So now given any

$$\array{ X && \underoverset{\in W}{f}{\longrightarrow} && Y \\ & {}_{\mathllap{\in Fib}}\searrow && \swarrow_{\mathrlap{\in Fib}} \\ && B }$$

apply the factorization lemma in $(\mathcal{C}_{/B})_f$ to factor it as

$$\array{ X &\overset{i \in W}{\longrightarrow}& Path_B(f) &\overset{\in W \cap Fib}{\longrightarrow}& Y \\ & {}_{\mathllap{\in Fib}}\searrow &\downarrow& \swarrow_{\mathrlap{\in Fib}} \\ && B } \,.$$

By the previous discussion it is sufficient now to show that the base change of $i$ to $B'$ is still a weak equivalence. But by the factorization lemma in $(\mathcal{C}_{/B})_f$, the morphism $i$ is right inverse to another acyclic fibration over $B$:

$$\array{ id_X \;\colon & X &\overset{i \in W}{\longrightarrow}& Path_B(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ & & {}_{\mathllap{\in Fib}}\searrow &\downarrow& \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,.$$

(Notice that if we had applied the factorization lemma of $\Delta_X$ in $\mathcal{C}_f$ instead of $(\Delta_X)/B$ in $(\mathcal{C}_{/B})$ then the corresponding triangle on the right here would not commute.)

Now we may reason as before: the base change of the top morphism here is exhibited by the following pasting composite of pullbacks:

$$\array{ B' \underset{B}{\times} X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ B' \underset{B}{\times} Path_B(f) &\longrightarrow& Path_B(f) \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in W \cap Fib}} \\ B' \underset{B}{\times}X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ B' &\longrightarrow& B } \,.$$

The acyclic fibration $Path_B(f)$ is preserved by this pullback, as is the identity $id_X \colon X \to Path_B(X)\to X$. Hence the weak equivalence $X \to Path_B(X)$ is preserved by two-out-of-three (def. ).

Lemma

In a category of fibrant objects, def. , the pullback of a weak equivalence along a fibration is again a weak equivalence.

(Brown 73, section 4, lemma 2)

Proof

Let $u \colon B' \to B$ be a weak equivalence and $p \colon E \to B$ be a fibration. We want to show that the left vertical morphism in the pullback

$$\array{ E \times_B B' &\longrightarrow& B' \\ \downarrow^{\mathrlap{\Rightarrow \in W} } && \;\downarrow^{\mathrlap{\in W}} \\ E &\stackrel{\in Fib}{\longrightarrow}& B }$$

is a weak equivalence.

First of all, using the factorization lemma we may factor $B' \to B$ as

$$B' \stackrel{\in W}{\longrightarrow} Path(u) \stackrel{\in W \cap F}{\longrightarrow} B$$

with the first morphism a weak equivalence that is a right inverse to an acyclic fibration and the right one an acyclic fibration.

Then the pullback diagram in question may be decomposed into two consecutive pullback diagrams

$$\array{ E \times_B B' &\to& B' \\ \downarrow && \downarrow \\ Q &\stackrel{\in Fib}{\to}& Path(u) \\ \;\;\downarrow^{\mathrlap{\in W \cap Fib}} && \;\;\downarrow^{\mathrlap{\in W \cap Fib}} \\ E &\stackrel{\in Fib}{\longrightarrow}& B } \,,$$

where the morphisms are indicated as fibrations and acyclic fibrations using the stability of these under arbitrary pullback.

This means that the proof reduces to proving that weak equivalences $u : B' \stackrel{\in W}{\to} B$ that are right inverse to some acyclic fibration $v : B \stackrel{\in W \cap F}{\to} B'$ map to a weak equivalence under pullback along a fibration.

Given such $u$ with right inverse $v$, consider the pullback diagram

$$\array{ & E \\ & {}^{\mathllap{{(p,id)}\atop \in W}}\downarrow & \searrow^{\mathrlap{id}} \\ E_1 \coloneqq & B \times_{B'} E & \stackrel{\in W \cap Fib }{\longrightarrow} & E \\ & \downarrow^{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p \in Fib }} \\ & &(pb)& B \\ & \downarrow && \downarrow^{\mathrlap{v \in W \cap Fib}} \\ & B &\overset{v \in Fib \cap W}{\longrightarrow}& B' } \,.$$

Notice that the indicated universal morphism $p \times Id \colon E \stackrel{\in W}{\to} E_1$ into the pullback is a weak equivalence by two-out-of-three (def. ).

The previous lemma says that weak equivalences between fibrations over $B$ are themselves preserved by base extension along $u \colon B' \to B$. In total this yields the following diagram

$$\array{ && u^* E = B' \times_B E &\longrightarrow & E \\ && {}^{\mathllap{ {u^*(p \times Id)} \atop {\in W} }}\downarrow && {}^{\mathllap{ {p \times Id} \atop {\in W} }}\downarrow & \searrow^{\mathrlap{id}} \\ && u^* E_1 &\longrightarrow& E_1 &\stackrel{\in W \cap Fib}{\longrightarrow}& E \\ &&\downarrow^{\mathrlap{\in Fib}}&&\downarrow^{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p \in Fib}} \\ &&&&&& B \\ &&\downarrow&&\downarrow && \downarrow^{\mathrlap{v \in W \cap Fib}} \\ && B' &\stackrel{u}{\longrightarrow}& B &\stackrel{v \in W \cap Fib}{\longrightarrow}& B' }$$

so that with $p \times Id : E \to E_1$ a weak equivalence also $u^* (p \times Id)$ is a weak equivalence, as indicated.

Notice that $u^* E = B' \times_B E \to E$ is the morphism that we want to show is a weak equivalence. By two-out-of-three (def. ) for that it is now sufficient to show that $u^* E_1 \to E_1$ is a weak equivalence.

That finally follows now since, by assumption, the total bottom horizontal morphism is the identity. Hence so is the top horizontal morphism. Therefore $u^\ast E_1 \to E_1$ is right inverse to a weak equivalence, hence is a weak equivalence.

Lemma

Let $(\mathcal{C}^{\ast/})_f$ be a category of fibrant objects, def. in a model structure on pointed objects (prop. ). Given any commuting diagram in $\mathcal{C}^{}$ of the form

$$\array{ X'_1 &\underoverset{t}{\in W}{\longrightarrow}& X_1 &\stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}& X_2 \\ && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in Fib}} \\ && B &\overset{u}{\longrightarrow}& C }$$

(meaning: both squares commute and $t$ equalizes $f$ with $g$) then the localization functor $\gamma \colon (\mathcal{C}^{\ast/})_f \to Ho(\mathcal{C}^{\ast/})$ (def. , cor ) takes the morphisms $fib(p_1) \stackrel{\longrightarrow}{\longrightarrow} fib(p_2)$ induced by $f$ and $g$ on fibers (example ) to the same morphism, in the homotopy category.

(Brown 73, section 4, lemma 4)

Proof

First consider the pullback of $p_2$ along $u$: this forms the same kind of diagram but with the bottom morphism an identity. Hence it is sufficient to consider this special case.

Consider the full subcategory $(\mathcal{C}^{\ast/}_{/B})_f$ of the slice category $\mathcal{C}^{\ast/}_{/B}$ (def. ) on its fibrant objects, i.e. the full subcategory of the slice category on the fibrations

$$\array{ X \\ \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib}} \\ B }$$

into $B$. By factorizing for every such fibration the diagonal morphisms into the fiber product $X \underset{B}{\times} X$ through a weak equivalence followed by a fibration, we obtain path space objects $Path_B(X)$ relative to $B$:

$$\array{ (\Delta_X)/B \;\colon & X &\overset{\in W}{\longrightarrow}& Path_B(X) &\overset{\in Fib}{\longrightarrow}& X \underset{B}{\times} X \\ & & {}_{\mathllap{\in Fib}}\searrow & \downarrow & \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,.$$

With these, the factorization lemma (lemma ) applies in $(\mathcal{C}^{\ast/}_{/B})_f$.

Let then $X\overset{s}{\to}Path_B(X_2)\overset{(p_0,p_1)}{\to} X_2 \times_B X_2$ be a path space object for $X_2$ in the slice over $B$ and consider the following commuting square

$$\array{ X'_1 &\overset{s f t}{\longrightarrow}& Path_B(X_2) \\ {}^{\mathllap{t}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{(p_0,p_1)}}_{\mathrlap{\in Fib}} \\ X_1 &\overset{(f,g)}{\longrightarrow}& X_2\underset{B}{\times} X_2 } \,.$$

By factoring this through the pullback $(f,g)^\ast(p_0,p_1)$ and then applying the factorization lemma and then two-out-of-three (def. ) to the factoring morphisms, this may be replaced by a commuting square of the same form, where however the left morphism is an acyclic fibration

$$\array{ X''_1 &\overset{}{\longrightarrow}& Path_B(X_2) \\ {}^{\mathllap{t}}_{\mathllap{\in W\cap Fib}} \downarrow && \downarrow^{\mathrlap{(p_0,p_1)}}_{\mathrlap{\in Fib}} \\ X_1 &\overset{(f,g)}{\longrightarrow}& X_2\underset{B}{\times} X_2 } \,.$$

This makes also the morphism $X''_1 \to B$ be a fibration, so that the whole diagram may now be regarded as a diagram in the category of fibrant objects $(\mathcal{C}_{/B})_f$ of the slice category over $B$.

As such, the top horizontal morphism now exhibits a right homotopy which under localization $\gamma_B \;\colon\; (\mathcal{C}_{/B})_f \longrightarrow Ho(\mathcal{C}_{/B})$ (def. ) of the slice model structure (prop. ) we have

$$\gamma_B(f) = \gamma_B(g) \,.$$

The result then follows by observing that we have a commuting square of functors

$$\array{ (\mathcal{C}^{\ast/}_{/B})_f &\overset{fib}{\longrightarrow}& \mathcal{C}^{\ast/} \\ \downarrow^{\mathrlap{\gamma_B}} &\swArrow& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}^{\ast/}_{/B}) &\longrightarrow& Ho(\mathcal{C}^{\ast/}) } \,,$$

because, by lemma , the top and right composite sends weak equivalences to isomorphisms, and hence the bottom filler exists by theorem . This implies the claim.

Homotopy fibers

We now discuss the homotopy-theoretic properties of the mapping cone- and mapping cocone-constructions from above.

Literature (Brown 73, section 4).

Remark

The factorization lemma with prop. says that the mapping cocone of a morphism $f$, def. , is equivalently the plain fiber, example , of a fibrant resolution $\tilde f$ of $f$:

$$\array{ Path_\ast(f) &\longrightarrow& Path(f) \\ \downarrow &(pb)& \downarrow^{\mathrlap{\tilde f}} \\ \ast &\longrightarrow& Y } \,.$$

The following prop. says that, up to equivalence, this situation is independent of the specific fibration resolution $\tilde f$ provided by the factorization lemma (hence by the prescription for the mapping cocone), but only depends on it being some fibration resolution.

Proposition

In the category of fibrant objects $(\mathcal{C}^{\ast/})_f$, def. , of a model structure on pointed objects (prop. ) consider a morphism of fiber-diagrams, hence a commuting diagram of the form

$$\array{ fib(p_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{p_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} \\ fib(p_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{p_2}{\longrightarrow}& Y_2 } \,.$$

If $f$ and $g$ weak equivalences, then so is $h$.

Proof

Factor the diagram in question through the pullback of $p_2$ along $f$

$$\array{ fib(p_1) &\longrightarrow& X_1 \\ \downarrow^{\mathrlap{h}} && {}^{\mathllap{\in W}}\downarrow & \searrow^{\mathrlap{p_1}} & \\ fib(f^\ast p_2) &\longrightarrow& f^\ast X_2 &\underoverset{\in Fib}{f^\ast p_2}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ fib(p_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{p_2}{\longrightarrow}& Y_2 }$$

and observe that

1. $fib(f^\ast p_2) = pt^\ast f^\ast p_2 = pt^\ast p_2 = fib(p_2)$;

2. $f^\ast X_2 \to X_2$ is a weak equivalence by lemma ;

3. $X_1 \to f^\ast X_2$ is a weak equivalence by assumption and by two-out-of-three (def. );

Moreover, this diagram exhibits $h \colon fib(p_1)\to fib(f^\ast p_2) = fib(p_2)$ as the base change, along $\ast \to Y_1$, of $X_1 \to f^\ast X_2$. Therefore the claim now follows with lemma .

Hence we say:

Definition

Let $\mathcal{C}$ be a model category and $\mathcal{C}^{\ast/}$ its model category of pointed objects, prop. . For $f \colon X \longrightarrow Y$ any morphism in its category of fibrant objects $(\mathcal{C}^{\ast/})_f$, def. , then its homotopy fiber

$$hofib(f)\longrightarrow X$$

is the morphism in the homotopy category $Ho(\mathcal{C}^{\ast/})$, def. , which is represented by the fiber, example , of any fibration resolution $\tilde f$ of $f$ (hence any fibration $\tilde f$ such that $f$ factors through a weak equivalence followed by $\tilde f$).

Dually:

For $f \colon X \longrightarrow Y$ any morphism in its category of cofibrant objects $(\mathcal{C}^{\ast/})_c$, def. , then its homotopy cofiber

$$Y \longrightarrow hocofib(f)$$

is the morphism in the homotopy category $Ho(\mathcal{C})$, def. , which is represented by the cofiber, example , of any cofibration resolution of $f$ (hence any cofibration $\tilde f$ such that $f$ factors as $\tilde f$ followed by a weak equivalence).

Proposition

The homotopy fiber in def. is indeed well defined, in that for $f_1$ and $f_2$ two fibration replacements of any morphisms $f$ in $\mathcal{C}_f$, then their fibers are isomorphic in $Ho(\mathcal{C}^{\ast/})$.

Proof

It is sufficient to exhibit an isomorphism in $Ho(\mathcal{C}^{\ast/})$ from the fiber of the fibration replacement given by the factorization lemma (for any choice of path space object) to the fiber of any other fibration resolution.

Hence given a morphism $f \colon Y \longrightarrow X$ and a factorization

$$f \;\colon\; X \underset{\in W}{\longrightarrow} \hat X \underoverset{f_1}{\in Fib}{\longrightarrow} Y$$

consider, for any choice $Path(Y)$ of path space object (def. ), the diagram

$$\array{ Path(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{\in W}}\downarrow &(pb)& \downarrow^{\mathrlap{\in W}} \\ Path(f_1) &\overset{\in W \cap Fib}{\longrightarrow}& \hat X \\ {}^{\mathllap{\in Fib}}\downarrow &(pb)& \downarrow^{\mathrlap{ {f_1} \atop {\in Fib}}} \\ Path(Y) &\underoverset{\in W \cap Fib}{p_1}{\longrightarrow}& Y \\ {}^{\mathllap{ {p_0} \atop \in W \cap Fib}}\downarrow \\ Y }$$

as in the proof of lemma . Now by repeatedly using prop. :

1. the bottom square gives a weak equivalence from the fiber of $Path(f_1) \to Path(Y)$ to the fiber of $f_1$;

2. The square

   $$\array{ Path(f_1) &\overset{id}{\longrightarrow}& Path(f_1) \\ \downarrow && \downarrow \\ Path(Y) &\underset{p_0}{\longrightarrow}& Y }$$

   gives a weak equivalence from the fiber of $Path(f_1) \to Path(Y)$ to the fiber of $Path(f_1)\to Y$.

3. Similarly the total vertical composite gives a weak equivalence via

   $$\array{ Path(f) &\overset{\in W}{\longrightarrow}& Path(f_1) \\ \downarrow && \downarrow \\ Y &\underset{id}{\longrightarrow}& Y }$$

from the fiber of $Path(f) \to Y$ to the fiber of $Path(f_1)\to Y$.

Together this is a zig-zag of weak equivalences of the form

$$fib(f_1) \;\overset{\in W}{\longleftarrow}\; fib(Path(f_1)\to Path(Y)) \;\overset{\in W}{\longrightarrow}\; fib(Path(f_1)\to Y) \;\overset{\in W}{\longleftarrow}\; fib(Path(f) \to Y)$$

between the fiber of $Path(f) \to Y$ and the fiber of $f_1$. This gives an isomorphism in the homotopy category.

Example

(fibers of Serre fibrations)

In showing that Serre fibrations are abstract fibrations in the sense of model category theory, theorem implies that the fiber $F$ (example ) of a Serre fibration, def.

$$\array{ F &\longrightarrow& X \\ && \downarrow^{\mathrlap{p}} \\ && B }$$

over any point is actually a homotopy fiber in the sense of def. . With prop. this implies that the weak homotopy type of the fiber only depends on the Serre fibration up to weak homotopy equivalence in that if $p' \colon X' \to B'$ is another Serre fibration fitting into a commuting diagram of the form

$$\array{ X &\overset{\in W_{cl}}{\longrightarrow}& X' \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{p'}} \\ B &\overset{\in W_{cl}}{\longrightarrow}& B' }$$

then $F \overset{\in W_{cl}}{\longrightarrow} F'$.

In particular this gives that the weak homotopy type of the fiber of a Serre fibration $p \colon X \to B$ does not change as the basepoint is moved in the same connected component. For let $\gamma \colon I \longrightarrow B$ be a path between two points

$$b_{0,1} \;\colon\; \ast \underoverset{\in W_{cl}}{i_{0,1}}{\longrightarrow} I \overset{\gamma}{\longrightarrow} B \,.$$

Then since all objects in $(Top_{cg})_{Quillen}$ are fibrant, and since the endpoint inclusions $i_{0,1}$ are weak equivalences, lemma gives the zig-zag of top horizontal weak equivalences in the following diagram:

$$\array{ F_{b_0} = & b_0^\ast p &\overset{\in W_{cl}}{\longrightarrow}& \gamma^{\ast}p &\overset{\in W_{cl}}{\longleftarrow}& b_1^\ast p & = F_{b_1} \\ & \downarrow &(pb)& \downarrow{\mathrlap{{\gamma^\ast f} \atop {\in \atop {Fib}}}} &\;\;(pb)& \downarrow \\ & \ast &\underoverset{i_0}{\in W_{cl}}{\longrightarrow}& I &\underoverset{i_1}{\in W_{cl}}{\longleftarrow}& \ast }$$

and hence an isomorphism $F_{b_0} \simeq F_{b_1}$ in the classical homotopy category (def. ).

The same kind of argument applied to maps from the square $I^2$ gives that if $\gamma_1, \gamma_2\colon I \to B$ are two homotopic paths with coinciding endpoints, then the isomorphisms between fibers over endpoints which they induce are equal. (But in general the isomorphism between the fibers does depend on the choice of homotopy class of paths connecting the basepoints!)

The same kind of argument also shows that if $B$ has the structure of a cell complex (def. ) then the restriction of the Serre fibration to one cell $D^n$ may be identified in the homotopy category with $D^n \times F$, and may be canonically identified so if the fundamental group of $X$ is trivial. This is used when deriving the Serre-Atiyah-Hirzebruch spectral sequence for $p$ (prop.).

Example

For every continuous function $f \colon X \longrightarrow Y$ between CW-complexes, def. , then the standard topological mapping cone is the attaching space (example )

$$Y \cup_f Cone(X) \;\; \in Top$$

of $Y$ with the standard cone $Cone(X)$ given by collapsing one end of the standard topological cyclinder $X \times I$ (def. ) as shown in example .

Equipped with the canonical continuous function

$$Y \longrightarrow Y \cup_f Cone(X)$$

this represents the homotopy cofiber, def. , of $f$ with respect to the classical model structure on topological spaces $\mathcal{C}= Top_{Quillen}$ from theorem .

Proof

By prop. , for $X$ a CW-complex then the standard topological cylinder object $X\times I$ is indeed a cyclinder object in $Top_{Quillen}$. Therefore by prop. and the factorization lemma , the mapping cone construction indeed produces first a cofibrant replacement of $f$ and then the ordinary cofiber of that, hence a model for the homotopy cofiber.

Example

The homotopy fiber of the inclusion of classifying spaces $B O(n) \hookrightarrow B O(n+1)$ is the n-sphere $S^n$. See this prop. at Classifying spaces and G-structure.

Example

Suppose a morphism $f \colon X \longrightarrow Y$ already happens to be a fibration between fibrant objects. The factorization lemma replaces it by a fibration out of the mapping cocylinder $Path(f)$, but such that the comparison morphism is a weak equivalence:

$$\array{ fib(f) &\longrightarrow& X &\underoverset{\in Fib}{f}{\longrightarrow}& Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{id}} \\ fib(\tilde f) &\longrightarrow& Path(f) &\underoverset{\in Fib}{\tilde f}{\longrightarrow}& Y } \,.$$

Hence by prop. in this case the ordinary fiber of $f$ is weakly equivalent to the mapping cocone, def. .

We may now state the abstract version of the statement of prop. :

Proposition

Let $\mathcal{C}$ be a model category. For $f \colon X \to Y$ any morphism of pointed objects, and for $A$ a pointed object, def. , then the sequence

$$[A,hofib(f)]_\ast \overset{i_\ast}{\longrightarrow} [A,X]_\ast \overset{f_\ast}{\longrightarrow} [A,Y]_{\ast}$$

is exact as a sequence of pointed sets.

(Where the sequence here is the image of the homotopy fiber sequence of def. under the hom-functor $[A,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/}$ from example .)

Proof

Let $A$, $X$ and $Y$ denote fibrant-cofibrant objects in $\mathcal{C}^{\ast/}$ representing the given objects of the same name in $Ho(\mathcal{C}^{\ast/})$. Moreover, let $f$ be a fibration in $\mathcal{C}^{\ast/}$ representing the given morphism of the same name in $Ho(\mathcal{C}^{\ast/})$.

Then by def. and prop. there is a representative $hofib(f) \in \mathcal{C}$ of the homotopy fiber which fits into a pullback diagram of the form

$$\array{ hofib(f) &\overset{i}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y }$$

With this the hom-sets in question are represented by genuine morphisms in $\mathcal{C}^{\ast/}$, modulo homotopy. From this it follows immediately that $im(i_\ast)$ includes into $ker(f_\ast)$. Hence it remains to show the converse: that every element in $ker(f_\ast)$ indeed comes from $im(i_\ast)$.

But an element in $ker(f_\ast)$ is represented by a morphism $\alpha \colon A \to X$ such that there is a left homotopy as in the following diagram

$$\array{ && A &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{i_0}}\downarrow &{}^{\tilde \eta}\nearrow& \downarrow^{\mathrlap{f}} \\ A &\overset{i_1}{\longrightarrow} & Cyl(A) &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow^{\mathrlap{=}} \\ \ast && \longrightarrow && Y } \,.$$

Now by lemma the square here has a lift $\tilde \eta$, as shown. This means that $i_1 \circ\tilde \eta$ is left homotopic to $\alpha$. But by the universal property of the fiber, $i_1 \circ \tilde \eta$ factors through $i \colon hofib(f) \to X$.

With prop. it also follows notably that the loop space construction becomes well-defined on the homotopy category:

Remark

Given an object $X \in \mathcal{C}^{\ast/}_f$, and picking any path space object $Path(X)$, def. with induced loop space object $\Omega X$, def. , write $Path_2(X) = Path(X) \underset{X}{\times} Path(X)$ for the path space object given by the fiber product of $Path(X)$ with itself, via example . From the pullback diagram there, the fiber inclusion $\Omega X \to Path(X)$ induces a morphism

$$\Omega X \times \Omega X \longrightarrow (\Omega X)_2 \,.$$

In the case where $\mathcal{C}^{\ast/} = Top^{\ast/}$ and $\Omega$ is induced, via def. , from the standard path space object (def. ), i.e. in the case that

$$\Omega X = fib(Maps(I_+,X)_\ast \longrightarrow X \times X) \,,$$

then this is the operation of concatenating two loops parameterized by $I = [0,1]$ to a single loop parameterized by $[0,2]$.

Proposition

Let $\mathcal{C}$ be a model category, def. . Then the construction of forming loop space objects $X\mapsto \Omega X$, def. (which on $\mathcal{C}^{\ast/}_f$ depends on a choice of path space objects, def. ) becomes unique up to isomorphism in the homotopy category (def. ) of the model structure on pointed objects (prop. ) and extends to a functor:

$$\Omega \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Ho(\mathcal{C}^{\ast/}) \,.$$

Dually, the reduced suspension operation, def. , which on $\mathcal{C}^{\ast/}$ depends on a choice of cylinder object, becomes a functor on the homotopy category

$$\Sigma \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Ho(\mathcal{C}^{\ast/}) \,.$$

Moreover, the pairing operation induced on the objects in the image of this functor via remark (concatenation of loops) gives the objects in the image of $\Omega$ group object structure, and makes this functor lift as

$$\Omega \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Grp(Ho(\mathcal{C}^{\ast/})) \,.$$

(Brown 73, section 4, theorem 3)

Proof

Given an object $X \in \mathcal{C}^{\ast/}$ and given two choices of path space objects $Path(X)$ and $\widetilde{Path(X)}$, we need to produce an isomorphism in $Ho(\mathcal{C}^{\ast/})$ between $\Omega X$ and $\tilde \Omega X$.

To that end, first lemma implies that any two choices of path space objects are connected via a third path space by a span of morphisms compatible with the structure maps. By two-out-of-three (def. ) every morphism of path space objects compatible with the inclusion of the base object is a weak equivalence. With this, lemma implies that these morphisms induce weak equivalences on the corresponding loop space objects. This shows that all choices of loop space objects become isomorphic in the homotopy category.

Moreover, all the isomorphisms produced this way are actually equal: this follows from lemma applied to

$$\array{ X &\overset{s}{\longrightarrow}& Path(X) &\stackrel{\longrightarrow}{\longrightarrow}& \widetilde{Path(X)} \\ && \downarrow && \downarrow \\ && X\times X &\overset{id}{\longrightarrow}& X \times X } \,.$$

This way we obtain a functor

$$\Omega \;\colon\; \mathcal{C}^{\ast/}_f \longrightarrow Ho(\mathcal{C}^{\ast/}) \,.$$

By prop. (and using that Cartesian product preserves weak equivalences) this functor sends weak equivalences to isomorphisms. Therefore the functor on homotopy categories now follows with theorem .

It is immediate to see that the operation of loop concatenation from remark gives the objects $\Omega X \in Ho(\mathcal{C}^{\ast/})$ the structure of monoids. It is now sufficient to see that these are in fact groups:

We claim that the inverse-assigning operation is given by the left map in the following pasting composite

$$\array{ \Omega' X &\longrightarrow& Path'(X) &\overset{}{\longrightarrow}& X \times X \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} &(pb)& \downarrow^{\mathrlap{swap}} \\ \Omega X &\longrightarrow& Path(X) &\underset{(p_0,p_1)}{\longrightarrow}& X \times X } \,,$$

(where $Path'(X)$, thus defined, is the path space object obtained from $Path(X)$ by “reversing the notion of source and target of a path”).

To see that this is indeed an inverse, it is sufficient to see that the two morphisms

$$\Omega X \stackrel{\longrightarrow}{\longrightarrow} (\Omega X)_2$$

induced from

$$\array{ Path(X) \stackrel{\overset{\Delta}{\longrightarrow}}{\underset{(s\circ p_0,s \circ p_0)}{\longrightarrow}} Path(X) \times_X Path'(X) }$$

coincide in the homotopy category. This follows with lemma applied to the following commuting diagram:

$$\array{ X &\overset{i}{\longrightarrow}& Path(X) &\stackrel{\overset{\Delta}{\longrightarrow}}{\underset{(s\circ p_0,s \circ p_0)}{\longrightarrow}}& Path(X)\times_X Path'(X) \\ && {}^{\mathllap{(p_0,p_1)}}\downarrow && \downarrow^{\mathrlap{}} \\ && X\times X &\overset{\Delta \circ pr_1}{\longrightarrow}& X \times X } \,.$$

Homotopy pullbacks

The concept of homotopy fibers of def. is a special case of the more general concept of homotopy pullbacks.

Definition

(proper model category)

A model category $\mathcal{C}$ (def. ) is called

• a right proper model category if pullback along fibrations preserves weak equivalences;

• a left proper model category if pushout along cofibrations preserves weak equivalences;

• a proper model category if it is both left and right proper.

Example

By lemma , a model category $\mathcal{C}$ (def. ) in which all objects are fibrant is a right proper model category (def. ).

Definition

Let $\mathcal{C}$ be a right proper model category (def. ). Then a commuting square

$$\array{ A &\longrightarrow& B \\ \downarrow && \downarrow^{\mathrlap{g}} \\ C &\underset{f}{\longrightarrow}& D }$$

in $\mathcal{C}_f$ is called a homotopy pullback (of $f$ along $g$ and equivalently of $g$ along $f$) if the following equivalent conditions hold:

1. for some factorization of the form

   $$g \colon B \overset{\in W }{\longrightarrow} \hat B \overset{\in Fib}{\longrightarrow} D$$

   the universally induced morphism from $A$ into the pullback of $\hat B$ along $f$ is a weak equivalence:

   $$\array{ A &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\longrightarrow& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\longrightarrow& D } \,.$$

2. for some factorization of the form

   $$f \colon C \overset{\in W }{\longrightarrow} \hat C \overset{\in Fib}{\longrightarrow} D$$

   the universally induced morphism from $A$ into the pullback of $\hat D$ along $g$ is a weak equivalence:

   $$A \overset{\in W}{\longrightarrow} \hat C \underset{D}{\times} B \,.$$

3. the above two conditions hold for every such factorization.

(e.g. Goerss-Jardine 96, II (8.14))

Proposition

The conditions in def. are indeed equivalent.

Proof

First assume that the first condition holds, in that

$$\array{ A &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\longrightarrow& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\longrightarrow& D } \,.$$

Then let

$$f \colon C \overset{\in W }{\longrightarrow} \hat C \overset{\in Fib}{\longrightarrow} D$$

be any factorization of $f$ and consider the pasting diagram (using the pasting law for pullbacks)

$$\array{ A &\overset{}{\longrightarrow}& \hat C \underset{D}{\times} B &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} &(pb)& \downarrow^{\mathrlap{\in W}} \\ C\underset{D}{\times} \hat B &\overset{\in W}{\longrightarrow}& \hat C \underset{D}{\times} \hat D &\overset{\in Fib}{\longrightarrow}& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in \atop Fib}} &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\underset{\in W}{\longrightarrow}& \hat C &\underset{\in Fib}{\longrightarrow}& D } \,,$$

where the inner morphisms are fibrations and weak equivalences, as shown, by the pullback stability of fibrations (prop. ) and then since pullback along fibrations preserves weak equivalences by assumption of right properness (def. ). Hence it follows by two-out-of-three (def. ) that also the comparison morphism $A \to \hat C \underset{D}{\times} B$ is a weak equivalence.

In conclusion, if the homotopy pullback condition is satisfied for one factorization of $g$, then it is satisfied for all factorizations of $f$. Since the argument is symmetric in $f$ and $g$, this proves the claim.

Remark

In particular, an ordinary pullback square of fibrant objects, one of whose edges is a fibration, is a homotopy pullback square according to def. .

Proposition

Let $\mathcal{C}$ be a right proper model category (def. ). Given a diagram in $\mathcal{C}$ of the form

$$\array{ A &\longrightarrow& B &\overset{\in Fib}{\longleftarrow}& C \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} \\ D &\longrightarrow& E &\underset{\in Fib}{\longleftarrow}& F }$$

then the induced morphism on pullbacks is a weak equivalence

$$A \underset{B}{\times} C \overset{\in W}{\longrightarrow} D \underset{E}{\times} F \,.$$

Proof

(The reader should draw the 3-dimensional cube diagram which we describe in words now.)

First consider the universal morphism $C \to E \underset{F}{\times} C$ and observe that it is a weak equivalence by right properness (def. ) and two-out-of-three (def. ).

Then consider the universal morphism $A \underset{B}{\times}C \to A \underset{B}{\times}(E \underset{F}{\times}C)$ and observe that this is also a weak equivalence, since $A \underset{B}{\times} C$ is the limiting cone of a homotopy pullback square by remark , and since the morphism is the comparison morphism to the pullback of the factorization constructed in the first step.

Now by using the pasting law, then the commutativity of the “left” face of the cube, then the pasting law again, one finds that $A \underset{B}{\times} (E \underset{F}{\times} C) \simeq A \underset{D}{\times} (D \underset{E} F{\times})$. Again by right properness this implies that $A \underset{B}{\times} (E \underset{F}{\times} C)\to D \underset{E}{\times} F$ is a weak equivalence.

With this the claim follows by two-out-of-three.

Homotopy pullbacks satisfy the usual abstract properties of pullbacks:

Proposition

Let $\mathcal{C}$ be a right proper model category (def. ). If in a commuting square in $\mathcal{C}$ one edge is a weak equivalence, then the square is a homotopy pullback square precisely if the opposite edge is a weak equivalence, too.

Proof

Consider a commuting square of the form

$$\array{ A &\longrightarrow& B \\ \downarrow && \downarrow \\ C &\underset{\in W}{\longrightarrow}& D } \,.$$

To detect whether this is a homotopy pullback, by def. and prop. , we are to choose any factorization of the right vertical morphism to obtain the pasting composite

$$\array{ A &\longrightarrow& B \\ \downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\overset{\in W}{\longrightarrow}& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\underset{\in W}{\longrightarrow}& D } \,.$$

Here the morphism in the middle is a weak equivalence by right properness (def. ). Hence it follows by two-out-of-three that the top left comparison morphism is a weak equivalence (and so the original square is a homotopy pullback) precisely if the top morphism is a weak equivalence.

Proposition

Let $\mathcal{C}$ be a right proper model category (def. ).

1. (pasting law) If in a commuting diagram

   $$\array{ A &\longrightarrow& B &\longrightarrow& C \\ \downarrow && \downarrow && \downarrow \\ D &\longrightarrow& E &\underset{}{\longrightarrow}& F }$$

   the square on the right is a homotoy pullback (def. ) then the left square is, too, precisely if the total rectangle is;

2. in the presence of functorial factorization (def. ) through weak equivalences followed by fibrations:

   every retract of a homotopy pullback square (in the category $\mathcal{C}_f^{\Box}$ of commuting squares in $\mathcal{C}_f$) is itself a homotopy pullback square.

Proof

For the first statement: choose a factorization of $C \overset{\in W}{\to} \hat F \overset{\in Fib}{\to} F$, pull it back to a factorization $B \to \hat B \overset{\in Fib}{\to} E$ and assume that $B \to \hat B$ is a weak equivalence, i.e. that the right square is a homotopy pullback. Now use the ordinary pasting law to conclude.

For the second statement: functorially choose a factorization of the two right vertical morphisms of the squares and factor the squares through the pullbacks of the corresponding fibrations along the bottom morphisms, respectively. Now the statement that the squares are homotopy pullbacks is equivalent to their top left vertical morphisms being weak equivalences. Factor these top left morphisms functorially as cofibrations followed by acyclic fibrations. Then the statement that the squares are homotopy pullbacks is equivalent to those top left cofibrations being acyclic. Now the claim follows using that the retract of an acyclic cofibration is an acyclic cofibration (prop. ).

$\,$

Long fiber sequences

The ordinary fiber, example , of a morphism has the property that taking it twice is always trivial:

$$\ast \simeq fib(fib(f)) \longrightarrow fib(f) \longrightarrow X \overset{f}{\longrightarrow} Y \,.$$

This is crucially different for the homotopy fiber, def. . Here we discuss how this comes about and what the consequences are.

Proposition

Let $\mathcal{C}_f$ be a category of fibrant objects of a model category, def. and let $f \colon X \longrightarrow Y$ be a morphism in its category of pointed objects, def. . Then the homotopy fiber of its homotopy fiber, def. , is isomorphic, in $Ho(\mathcal{C}^{\ast/})$, to the loop space object $\Omega Y$ of $Y$ (def. , prop. ):

$$hofib(hofib(X \overset{f}{\to}Y)) \simeq \Omega Y \,.$$

Proof

Assume without restriction that $f \;\colon\; X \longrightarrow Y$ is already a fibration between fibrant objects in $\mathcal{C}$ (otherwise replace and rename). Then its homotopy fiber is its ordinary fiber, sitting in a pullback square

$$\array{ hofib(f) \simeq & F &\overset{i}{\longrightarrow}& X \\ & \downarrow && \downarrow^{\mathrlap{f}} \\ & \ast &\longrightarrow& Y } \,.$$

In order to compute $hofib(hofib(f))$, i.e. $hofib(i)$, we need to replace the fiber inclusion $i$ by a fibration. Using the factorization lemma for this purpose yields, after a choice of path space object $Path(X)$ (def. ), a replacement of the form

$$\array{ F &\overset{\in W}{\longrightarrow}& F \times_X Path(X) \\ &{}_{\mathllap{i}}\searrow& \downarrow^{\mathrlap{\tilde i}}_{\mathrlap{\in Fib}} \\ && X } \,.$$

Hence $hofib(i)$ is the ordinary fiber of this map:

$$hofib(hofib(f)) \simeq F \times_X Path(X) \times_X \ast \;\;\;\; \in Ho(\mathcal{C}^{\ast/}) \,.$$

Notice that

$$F \times_X Path(X) \; \simeq \; \ast \times_Y Path(X)$$

because of the pasting law:

$$\array{ F \times_X Path(X) &\longrightarrow& Path(X) \\ \downarrow &(pb)& \downarrow \\ F &\overset{i}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } \,.$$

Hence

$$hofib(hofib(f)) \;\simeq\; \ast \times_Y Path(X) \times_X \ast \,.$$

Now we claim that there is a choice of path space objects $Path(X)$ and $Path(Y)$ such that this model for the homotopy fiber (as an object in $\mathcal{C}^{\ast/}$) sits in a pullback diagram of the following form:

$$\array{ \ast \times_Y Path(X) \times_X \ast &\longrightarrow& Path(X) \\ \downarrow && \downarrow\mathrlap{\in W \cap F} \\ \Omega Y &\longrightarrow& Path(Y)\times_Y X \\ \downarrow &(pb)& \downarrow \\ \ast &\longrightarrow& Y \times X } \,.$$

By the pasting law and the pullback stability of acyclic fibrations, this will prove the claim.

To see that the bottom square here is indeed a pullback, check the universal property: A morphism out of any $A$ into $\ast \underset{Y \times X}{\times} Path(Y) \times_Y X$ is a morphism $a \colon A \to Path(Y)$ and a morphism $b \colon A \to X$ such that $p_0(a) = \ast$, $p_1(a) = f(b)$ and $b = \ast$. Hence it is equivalently just a morphism $a \colon A \to Path(Y)$ such that $p_0(a) = \ast$ and $p_1(a) = \ast$. This is the defining universal property of $\Omega Y \coloneqq \ast \underset{Y}{\times} Path(Y) \underset{Y}{\times} \ast$.

Now to construct the right vertical morphism in the top square (Quillen 67, page 3.1): Let $Path(Y)$ be any path space object for $Y$ and let $Path(X)$ be given by a factorization

$$(id_X, \; i \circ f, \; id_X) \;\colon\; X \overset{\in W}{\to} Path(X) \overset{\in Fib}{\longrightarrow} X \times_Y Path(Y) \times_Y X$$

and regarded as a path space object of $X$ by further comoposing with

$$(pr_1,pr_3)\colon X \times_Y Path(Y) \times_Y X \overset{\in Fib}{\longrightarrow} X \times X \,.$$

We need to show that $Path(X)\to Path(Y) \times_Y X$ is an acyclic fibration.

It is a fibration because $X\times_Y Path(Y) \times_Y X \to Path(Y)\times_Y X$ is a fibration, this being the pullback of the fibration $X \overset{f}{\to} Y$.

To see that it is also a weak equivalence, first observe that $Path(Y)\times_Y X \overset{\in W \cap Fib}{\longrightarrow} X$, this being the pullback of the acyclic fibration of lemma . Hence we have a factorization of the identity as

$$id_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{}{\longrightarrow} Path(Y)\times_Y X \underset{\in W \cap Fib}{\longrightarrow} X$$

and so finally the claim follows by two-out-of-three (def. ).

Remark

There is a conceptual way to understand prop. as follows: If we draw double arrows to indicate homotopies, then a homotopy fiber (def. ) is depicted by the following filled square:

$$\array{ hofib(f) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ X &\underset{f}{\longrightarrow}& Y }$$

just like the ordinary fiber (example ) is given by a plain square

$$\array{ fib(f) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ X &\underset{f}{\longrightarrow}& Y } \,.$$

One may show that just like the fiber is the universal solution to making such a commuting square (a pullback limit cone def. ), so the homotopy fiber is the universal solution up to homotopy to make such a commuting square up to homotopy – a homotopy pullback homotopy limit cone.

Now just like ordinary pullbacks satisfy the pasting law saying that attaching two pullback squares gives a pullback rectangle, the analogue is true for homotopy pullbacks. This implies that if we take the homotopy fiber of a homotopy fiber, thereby producing this double homotopy pullback square

$$\array{ hofib(g) &\longrightarrow& hofib(f) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow^{\mathrlap{g}} &\swArrow& \downarrow \\ \ast &\longrightarrow& X &\underset{f}{\longrightarrow}& Y }$$

then the total outer rectangle here is itself a homotopy pullback. But the outer rectangle exhibits the homotopy fiber of the point inclusion, which, via def. and lemma , is the loop space object:

$$\array{ \Omega Y &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ \ast &\longrightarrow& Y } \,.$$

Proposition

(long homotopy fiber sequences)

Let $\mathcal{C}$ be a model category and let $f \colon X \to Y$ be morphism in the pointed homotopy category $Ho(\mathcal{C}^{\ast/})$ (prop. ). Then:

1. There is a long sequence to the left in $\mathcal{C}^{\ast/}$ of the form

   $$\cdots \longrightarrow \Omega X \overset{\overline{\Omega} f}{\longrightarrow} \Omega Y \longrightarrow hofib(f) \longrightarrow X \overset{f}{\longrightarrow} Y \,,$$

   where each morphism is the homotopy fiber (def. ) of the following one: the homotopy fiber sequence of $f$. Here $\overline{\Omega}f$ denotes $\Omega f$ followed by forming inverses with respect to the group structure on $\Omega(-)$ from prop. .

   Moreover, for $A\in \mathcal{C}^{\ast/}$ any object, then there is a long exact sequence

   $$\cdots \to [A,\Omega^2 Y]_\ast \longrightarrow [A,\Omega hofib(f)]_\ast \longrightarrow [A, \Omega X]_\ast \longrightarrow [A,\Omega Y] \longrightarrow [A,hofib(f)]_\ast \longrightarrow [A,X]_\ast \longrightarrow [A,Y]_\ast$$

   of pointed sets, where $[-,-]_\ast$ denotes the pointed set valued hom-functor of example .

2. Dually, there is a long sequence to the right in $\mathcal{C}^{\ast/}$ of the form

   $$X \overset{f}{\longrightarrow} Y \overset{}{\longrightarrow} hocofib(f) \longrightarrow \Sigma X \overset{\overline{\Sigma} f}{\longrightarrow} \Sigma Y \to \cdots \,,$$

   where each morphism is the homotopy cofiber (def. ) of the previous one: the homotopy cofiber sequence of $f$. Moreover, for $A\in \mathcal{C}^{\ast/}$ any object, then there is a long exact sequence

   $$\cdots \to [\Sigma^2 X, A]_\ast \longrightarrow [\Sigma hocofib(f), A]_\ast \longrightarrow [\Sigma Y, A]_\ast \longrightarrow [\Sigma X, A] \longrightarrow [hocofib(f),A]_\ast \longrightarrow [Y,A]_\ast \longrightarrow [X,A]_\ast$$

   of pointed sets, where $[-,-]_\ast$ denotes the pointed set valued hom-functor of example .

(Quillen 67, I.3, prop. 4)

Proof

That there are long sequences of this form is the result of combining prop. and prop. .

It only remains to see that it is indeed the morphisms $\overline{\Omega} f$ that appear, as indicated.

In order to see this, it is convenient to adopt the following notation: for $f \colon X \to Y$ a morphism, then we denote the collection of generalized element of its homotopy fiber as

$$hofib(f) = \left\{ (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast) \right\}$$

indicating that these elements are pairs consisting of an element $x$ of $X$ and a “path” (an element of the given path space object) from $f(x)$ to the basepoint.

This way the canonical map $hofib(f) \to X$ is $(x, f(x) \rightsquigarrow \ast) \mapsto x$. Hence in this notation the homotopy fiber of the homotopy fiber reads

$$hofib(hofib(f)) = \left\{ ( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ) \right\} \,.$$

This identifies with $\Omega Y$ by forming the loops

$$\gamma_1 \cdot f(\overline{\gamma_2}) \,,$$

where the overline denotes reversal and the dot denotes concatenation.

Then consider the next homotopy fiber

$$hofib(hofib(hofib(f))) = \left\{ \left( ( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ), \left( \array{ x && \overset{\gamma_3}{\rightsquigarrow} && \ast \\ f(x) &&\overset{f(\gamma_3)}{\rightsquigarrow}&& \ast \\ & {}_{\mathllap{\gamma_1}}\searrow & \Rightarrow & \swarrow_{\mathllap{}} \\ && \ast } \right) \right) \right\} \,,$$

where on the right we have a path in $hofib(f)$ from $(x, f(x)\overset{\gamma_1}{\rightsquigarrow} \ast)$ to the basepoint element. This is a path $\gamma_3$ together with a path-of-paths which connects $f_1$ to $f(\gamma_3)$.

By the above convention this is identified with the loop in $X$ which is

$$\gamma_2 \cdot (\overline{\gamma}_3) \,.$$

But the map to $hofib(hofib(f))$ sends this data to $( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast )$, hence to the loop

$$\begin{aligned} \gamma_1 \cdot f( \overline{\gamma_2} ) & \simeq f(\gamma_3) \cdot f(\overline{\gamma_2}) \\ & = f( \gamma_3 \cdot \overline{\gamma_2} ) \\ & = f ( \overline{\gamma_2 \cdot \overline{\gamma}_3} ) \\ & = \overline{f(\gamma_2 \cdot \overline{\gamma}_3) } \end{aligned} \,,$$

hence to the reveral of the image under $f$ of the loop in $X$.

Remark

In (Quillen 67, I.3, prop. 3, prop. 4) more is shown than stated in prop. : there the connecting homomorphism $\Omega Y \to hofib(f)$ is not just shown to exist, but is described in detail via an action of $\Omega Y$ on $hofib(f)$ in $Ho(\mathcal{C})$. This takes a good bit more work. For our purposes here, however, it is sufficient to know that such a morphism exists at all, hence that $\Omega Y \simeq hofib(hofib(f))$.

Example

Let $\mathcal{C} = (Top_{cg})_{Quillen}$ be the classical model structure on topological spaces (compactly generated) from theorem , theorem . Then using the standard pointed topological path space objects $Maps(I_+,X)$ from def. and example as the abstract path space objects in def. , via prop. , this gives that

$$[\ast, \Omega^n X] \simeq \pi_n(X)$$

is the $n$th homotopy group, def. , of $X$ at its basepoint.

Hence using $A = \ast$ in the first item of prop. , the long exact sequence this gives is of the form

$$\cdots \to \pi_3(X) \overset{f_\ast}{\longrightarrow} \pi_3(Y) \longrightarrow \pi_2(hofib(f)) \overset{}{\longrightarrow} \pi_2(X) \overset{-f_\ast}{\longrightarrow} \pi_2(Y) \longrightarrow \pi_1(hofib(f)) \overset{}{\longrightarrow} \pi_1(X) \overset{f_\ast}{\longrightarrow} \pi_1(Y) \overset{}{\longrightarrow} \ast \,.$$

This is called the long exact sequence of homotopy groups induced by $f$.

Remark

As we pass to stable homotopy theory (in Part 1)), the long exact sequences in example become long not just to the left, but also to the right. Given then a tower of fibrations, there is an induced sequence of such long exact sequences of homotopy groups, which organizes into an exact couple. For more on this see at Interlude – Spectral sequences (this remark).

Example

Let again $\mathcal{C} = (Top_{cg})_{Quillen}$ be the classical model structure on topological spaces (compactly generated) from theorem , theorem , as in example . For $E \in Top_{cg}^{\ast/}$ any pointed topological space and $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, the exactness of the sequence in the second item of prop.

$$\cdots \to [hocofib(i), E] \longrightarrow [X,E]_\ast \longrightarrow [A,E]_\ast \to \cdots$$

gives that the functor

$$[-,E]_\ast \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Set^{\ast/}$$

behaves like one degree in an additive reduced cohomology theory (def.). The Brown representability theorem (thm.) implies that all additive reduced cohomology theories are degreewise representable this way (prop.).

$\,$

$\infty$-Groupoids I): Topological homotopy theory

This section first recalls relevant concepts from actual topology (“point-set topology”) and highlights facts that motivate the axiomatics of model categories below. We prove two technical lemmas (lemma and lemma ) that serve to establish the abstract homotopy theory of topological spaces further below.

Then we discuss how the category Top of topological spaces satisfies the axioms of abstract homotopy theory (model category) theory, def. .

Literature (Hirschhorn 15)

$\,$

Throughout, let Top denote the category whose objects are topological spaces and whose morphisms are continuous functions between them. Its isomorphisms are the homeomorphisms.

(Further below we restrict attention to the full subcategory of compactly generated topological spaces.)

Universal constructions

To begin with, we recall some basics on universal constructions in Top: limits and colimits of diagrams of topological spaces; exponential objects.

We now discuss limits and colimits (Def. ) in $\mathcal{C}=$ Top. The key for understanding these is the fact that there are initial and final topologies:

Definition

Let $\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}$ be a set of topological spaces, and let $S \in Set$ be a bare set. Then

1. For $\{S \stackrel{f_i}{\to} S_i \}_{i \in I}$ a set of functions out of $S$, the initial topology $\tau_{initial}(\{f_i\}_{i \in I})$ is the topology on $S$ with the minimum collection of open subsets such that all $f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i$ are continuous.

2. For $\{S_i \stackrel{f_i}{\to} S\}_{i \in I}$ a set of functions into $S$, the final topology $\tau_{final}(\{f_i\}_{i \in I})$ is the topology on $S$ with the maximum collection of open subsets such that all $f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))$ are continuous.

Example

For $X$ a single topological space, and $\iota_S \colon S \hookrightarrow U(X)$ a subset of its underlying set, then the initial topology $\tau_{intial}(\iota_S)$, def. , is the subspace topology, making

$$\iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X$$

a topological subspace inclusion.

Example

Conversely, for $p_S \colon U(X) \longrightarrow S$ an epimorphism, then the final topology $\tau_{final}(p_S)$ on $S$ is the quotient topology.

Proposition

Let $I$ be a small category and let $X_\bullet \colon I \longrightarrow Top$ be an $I$-diagram in Top (a functor from $I$ to $Top$), with components denoted $X_i = (S_i, \tau_i)$, where $S_i \in Set$ and $\tau_i$ a topology on $S_i$. Then:

1. The limit of $X_\bullet$ exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. , for the functions $p_i$ which are the limiting cone components:

   $$\array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.$$

   Hence

   $$\underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$$

2. The colimit of $X_\bullet$ exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology, def. for the component maps $\iota_i$ of the colimiting cocone

   $$\array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.$$

   Hence

   $$\underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)$$

(e.g. Bourbaki 71, section I.4)

Proof

The required universal property of $\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$ (def. ) is immediate: for

$$\array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j }$$

any cone over the diagram, then by construction there is a unique function of underlying sets $S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i$ making the required diagrams commute, and so all that is required is that this unique function is always continuous. But this is precisely what the initial topology ensures.

The case of the colimit is formally dual.

Example

The limit over the empty diagram in $Top$ is the point $\ast$ with its unique topology.

Example

For $\{X_i\}_{i \in I}$ a set of topological spaces, their coproduct $\underset{i \in I}{\sqcup} X_i \in Top$ is their disjoint union.

In particular:

Example

For $S \in Set$, the $S$-indexed coproduct of the point, $\underset{s \in S}{\coprod}\ast$ is the set $S$ itself equipped with the final topology, hence is the discrete topological space on $S$.

Example

For $\{X_i\}_{i \in I}$ a set of topological spaces, their product $\underset{i \in I}{\prod} X_i \in Top$ is the Cartesian product of the underlying sets equipped with the product topology, also called the Tychonoff product.

In the case that $S$ is a finite set, such as for binary product spaces $X \times Y$, then a sub-basis for the product topology is given by the Cartesian products of the open subsets of (a basis for) each factor space.

Example

The equalizer of two continuous functions $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the equalizer of the underlying functions of sets

$$eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y$$

(hence the largets subset of $S_X$ on which both functions coincide) and equipped with the subspace topology, example .

Example

The coequalizer of two continuous functions $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the coequalizer of the underlying functions of sets

$$S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g)$$

(hence the quotient set by the equivalence relation generated by $f(x) \sim g(x)$ for all $x \in X$) and equipped with the quotient topology, example .

Example

For

$$\array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X }$$

two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted

$$\array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,.$$

(Here $g_\ast f$ is also called the pushout of $f$, or the cobase change of $f$ along $g$.)

This is equivalently the coequalizer of the two morphisms from $A$ to the coproduct of $X$ with $Y$ (example ):

$$A \stackrel{\longrightarrow}{\longrightarrow} X \sqcup Y \longrightarrow X \sqcup_A Y \,.$$

If $g$ is an inclusion, one also writes $X \cup_f Y$ and calls this the attaching space.

By example the pushout/attaching space is the quotient topological space

$$X \sqcup_A Y \simeq (X\sqcup Y)/\sim$$

of the disjoint union of $X$ and $Y$ subject to the equivalence relation which identifies a point in $X$ with a point in $Y$ if they have the same pre-image in $A$.

(graphics from Aguilar-Gitler-Prieto 02)

Notice that the defining universal property of this colimit means that completing the span

$$\array{ A &\longrightarrow& Y \\ \downarrow \\ X }$$

to a commuting square

$$\array{ A &\longrightarrow& Y \\ \downarrow && \downarrow \\ X &\longrightarrow& Z }$$

is equivalent to finding a morphism

$$X \underset{A}{\sqcup} Y \longrightarrow Z \,.$$

Example

For $A\hookrightarrow X$ a topological subspace inclusion, example , then the pushout

$$\array{ A &\hookrightarrow& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A }$$

is the quotient space or cofiber, denoted $X/A$.

Example

An important special case of example :

For $n \in \mathbb{N}$ write

• $D^n \coloneqq \{ \vec x\in \mathbb{R}^n | \; {\vert \vec x \vert \leq 1}\} \hookrightarrow \mathbb{R}^n$ for the standard topological n-disk (equipped with its subspace topology as a subset of Cartesian space);

• $S^{n-1} = \partial D^n \coloneqq \{ \vec x\in \mathbb{R}^n | \; {\vert \vec x \vert = 1}\} \hookrightarrow \mathbb{R}^n$ for its boundary, the standard topological n-sphere.

Notice that $S^{-1} = \emptyset$ and that $S^0 = \ast \sqcup \ast$.

Let

$$i_n \colon S^{n-1}\longrightarrow D^n$$

be the canonical inclusion of the standard (n-1)-sphere as the boundary of the standard n-disk (both regarded as topological spaces with their subspace topology as subspaces of the Cartesian space $\mathbb{R}^n$).

Then the colimit in Top under the diagram

$$D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \,,$$

i.e. the pushout of $i_n$ along itself, is the n-sphere $S^n$:

$$\array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,.$$

(graphics from Ueno-Shiga-Morita 95)

Another kind of colimit that will play a role for certain technical constructions is transfinite composition. First recall

Definition

A partial order is a set $S$ equipped with a relation $\leq$ such that for all elements $a,b,c \in S$

1) (reflexivity) $a \leq a$;

2) (transitivity) if $a \leq b$ and $b \leq c$ then $a \leq c$;

3) (antisymmetry) if $a\leq b$ and $\b \leq a$ then $a = b$.

This we may and will equivalently think of as a category with objects the elements of $S$ and a unique morphism $a \to b$ precisely if $a\leq b$. In particular an order-preserving function between partially ordered sets is equivalently a functor between their corresponding categories.

A bottom element $\bot$ in a partial order is one such that $\bot \leq a$ for all a. A top element $\top$ is one for wich $a \leq \top$.

A partial order is a total order if in addition

4) (totality) either $a\leq b$ or $b \leq a$.

A total order is a well order if in addition

5) (well-foundedness) every non-empty subset has a least element.

An ordinal is the equivalence class of a well-order.

The successor of an ordinal is the class of the well-order with a top element freely adjoined.

A limit ordinal is one that is not a successor.

Example

The finite ordinals are labeled by $n \in \mathbb{N}$, corresponding to the well-orders $\{0 \leq 1 \leq 2 \cdots \leq n-1\}$. Here $(n+1)$ is the successor of $n$. The first non-empty limit ordinal is $\omega = [(\mathbb{N}, \leq)]$.

Definition

Let $\mathcal{C}$ be a category, and let $I \subset Mor(\mathcal{C})$ be a class of its morphisms.

For $\alpha$ an ordinal (regarded as a category), an $\alpha$-indexed transfinite sequence of elements in $I$ is a diagram

$$X_\bullet \;\colon\; \alpha \longrightarrow \mathcal{C}$$

such that

1. $X_\bullet$ takes all successor morphisms $\beta \stackrel{\leq}{\to} \beta + 1$ in $\alpha$ to elements in $I$

   $$X_{\beta,\beta + 1} \in I$$

2. $X_\bullet$ is continuous in that for every nonzero limit ordinal $\beta \lt \alpha$, $X_\bullet$ restricted to the full-subdiagram $\{\gamma \;|\; \gamma \leq \beta\}$ is a colimiting cocone in $\mathcal{C}$ for $X_\bullet$ restricted to $\{\gamma \;|\; \gamma \lt \beta\}$.

The corresponding transfinite composition is the induced morphism

$$X_0 \longrightarrow X_\alpha \coloneqq \underset{\longrightarrow}{\lim}X_\bullet$$

into the colimit of the diagram, schematically:

$$\array{ X_0 &\stackrel{X_{0,1}}{\to}& X_1 &\stackrel{X_{1,2}}{\to}& X_2 &\to& \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ && X_\alpha } \,.$$

We now turn to the discussion of mapping spaces/exponential objects.

Definition

For $X$ a topological space and $Y$ a locally compact topological space (in that for every point, every neighbourhood contains a compact neighbourhood), the mapping space

$$X^Y \in Top$$

is the topological space

• whose underlying set is the set $Hom_{Top}(Y,X)$ of continuous functions $Y \to X$,

• whose open subsets are unions of finitary intersections of the following subbase elements of standard open subsets:

  the standard open subset $U^K \subset Hom_{Top}(Y,X)$ for

  • $K \hookrightarrow Y$ a compact topological space subset

  • $U \hookrightarrow X$ an open subset

  is the subset of all those continuous functions $f$ that fit into a commuting diagram of the form

  $$\array{ K &\hookrightarrow& Y \\ \downarrow && \downarrow^{\mathrlap{f}} \\ U &\hookrightarrow& X } \,.$$

Accordingly this is called the compact-open topology on the set of functions.

The construction extends to a functor

$$(-)^{(-)} \;\colon\; Top_{lc}^{op} \times Top \longrightarrow Top \,.$$

Proposition

For $X$ a topological space and $Y$ a locally compact topological space (in that for each point, each open neighbourhood contains a compact neighbourhood), the topological mapping space $X^Y$ from def. is an exponential object, i.e. the functor $(-)^Y$ is right adjoint to the product functor $Y \times (-)$: there is a natural bijection

$$Hom_{Top}(Z \times Y, X) \simeq Hom_{Top}(Z, X^Y)$$

between continuous functions out of any product topological space of $Y$ with any $Z \in Top$ and continuous functions from $Z$ into the mapping space.

A proof is spelled out here (or see e.g. Aguilar-Gitler-Prieto 02, prop. 1.3.1).

Remark

In the context of prop. it is often assumed that $Y$ is also a Hausdorff topological space. But this is not necessary. What assuming Hausdorffness only achieves is that all alternative definitions of “locally compact” become equivalent to the one that is needed for the proposition: for every point, every open neighbourhood contains a compact neighbourhood.

Remark

Proposition fails in general if $Y$ is not locally compact. Therefore the plain category Top of all topological spaces is not a Cartesian closed category.

This is no problem for the construction of the homotopy theory of topological spaces as such, but it becomes a technical nuisance for various constructions that one would like to perform within that homotopy theory. For instance on general pointed topological spaces the smash product is in general not associative.

On the other hand, without changing any of the following discussion one may just pass to a more convenient category of topological spaces such as notably the full subcategory of compactly generated topological spaces $Top_{cg} \hookrightarrow Top$ (def. ) which is Cartesian closed. This we turn to below.

Homotopy

The fundamental concept of homotopy theory is clearly that of homotopy. In the context of topological spaces this is about contiunous deformations of continuous functions parameterized by the standard interval:

Definition

Write

$$I \coloneqq [0,1] \hookrightarrow \mathbb{R}$$

for the standard topological interval, a compact connected topological subspace of the real line.

Equipped with the canonical inclusion of its two endpoints

$$\ast \sqcup \ast \stackrel{(\delta_0,\delta_1)}{\longrightarrow} I \stackrel{\exists !}{\longrightarrow} \ast$$

this is the standard interval object in Top.

For $X \in Top$, the product topological space $X\times I$, example , is called the standard cylinder object over $X$. The endpoint inclusions of the interval make it factor the codiagonal on $X$

$$\nabla_X \;\colon\; X \sqcup X \stackrel{((id,\delta_0),(id,\delta_1))}{\longrightarrow} X \times I \longrightarrow X \,.$$

Definition

(left homotopy)

For $f,g\colon X \longrightarrow Y$ two continuous functions between topological spaces $X,Y$, then a left homotopy

$$\eta \colon f \,\Rightarrow_L\, g$$

is a continuous function

$$\eta \;\colon\; X \times I \longrightarrow Y$$

out of the standard cylinder object over $X$, def. , such that this fits into a commuting diagram of the form

$$\array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.$$

(graphics grabbed from J. Tauber here)

Example

Let $X$ be a topological space and let $x,y \in X$ be two of its points, regarded as functions $x,y \colon \ast \longrightarrow X$ from the point to $X$. Then a left homotopy, def. , between these two functions is a commuting diagram of the form

$$\array{ \ast \\ {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}} \\ I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}} \\ \ast } \,.$$

This is simply a continuous path in $X$ whose endpoints are $x$ and $y$.

For instance:

Example

Let

$$const_0 \;\colon\; I \longrightarrow \ast \overset{\delta_0}{\longrightarrow} I$$

be the continuous function from the standard interval $I = [0,1]$ to itself that is constant on the value 0. Then there is a left homotopy, def. , from the identity function

$$\eta \;\colon\; id_I \Rightarrow const_0$$

given by

$$\eta(x,t) \coloneqq x(1-t) \,.$$

A key application of the concept of left homotopy is to the definition of homotopy groups:

Definition

For $X$ a topological space, then its set $\pi_0(X)$ of connected components, also called the 0-th homotopy set, is the set of left homotopy-equivalence classes (def. ) of points $x \colon \ast \to X$, hence the set of path-connected components of $X$ (example ). By composition this extends to a functor

$$\pi_0 \colon Top \longrightarrow Set \,.$$

For $n \in \mathbb{N}$, $n \geq 1$ and for $x \colon \ast \to X$ any point, then the $n$th homotopy group $\pi_n(X,x)$ of $X$ at $x$ is the group

• whose underlying set is the set of left homotopy-equivalence classes of maps $I^n \longrightarrow X$ that take the boundary of $I^n$ to $x$ and where the left homotopies $\eta$ are constrained to be constant on the boundary;

• whose group product operation takes $[\alpha \colon I^n \to X]$ and $[\beta \colon I^n \to X]$ to $[\alpha \cdot \beta]$ with

$$\alpha \cdot \beta \;\colon\; I^n \stackrel{\simeq}{\longrightarrow} I^n \underset{I^{n-1}}{\sqcup} I^n \stackrel{(\alpha,\beta)}{\longrightarrow} X \,,$$

where the first map is a homeomorphism from the unit $n$-cube to the $n$-cube with one side twice the unit length (e.g. $(x_1, x_2, x_3, \cdots) \mapsto (2 x_1, x_2, x_3, \cdots)$).

By composition, this construction extends to a functor

$$\pi_{\bullet \geq 1} \;\colon\; Top^{\ast/} \longrightarrow Grp^{\mathbb{N}_{\geq 1}}$$

from pointed topological spaces to graded groups.

Notice that often one writes the value of this functor on a morphism $f$ as $f_\ast = \pi_\bullet(f)$.

Remark

At this point we don’t go further into the abstract reason why def. yields group structure above degree 0, which is that positive dimension spheres are H-cogroup objects. But this is important, for instance in the proof of the Brown representability theorem. See the section Brown representability theorem in Part S.

Definition

(homotopy equivalence)

A continuous function $f \;\colon\; X \longrightarrow Y$ is called a homotopy equivalence if there exists a continuous function the other way around, $g \;\colon\; Y \longrightarrow X$, and left homotopies, def. , from the two composites to the identity:

$$\eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y$$

and

$$\eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,.$$

If here $\eta_2$ is constant along $I$, $f$ is said to exhibit $X$ as a deformation retract of $Y$.

Example

For $X$ a topological space and $X \times I$ its standard cylinder object of def. , then the projection $p \colon X \times I \longrightarrow X$ and the inclusion $(id, \delta_0) \colon X \longrightarrow X\times I$ are homotopy equivalences, def. , and in fact are homotopy inverses to each other:

The composition

$$X \overset{(id,\delta_0)}{\longrightarrow} X\times I \overset{p}{\longrightarrow} X$$

is immediately the identity on $X$ (i.e. homotopic to the identity by a trivial homotopy), while the composite

$$X \times I \overset{p}{\longrightarrow} X \overset{(id, \delta_0)}{\longrightarrow} X\times I$$

is homotopic to the identity on $X \times I$ by a homotopy that is pointwise in $X$ that of example .

Definition

A continuous function $f \colon X \longrightarrow Y$ is called a weak homotopy equivalence if its image under all the homotopy group functors of def. is an isomorphism, hence if

$$\pi_0(f) \;\colon\; \pi_0(X) \stackrel{\simeq}{\longrightarrow} \pi_0(X)$$

and for all $x \in X$ and all $n \geq 1$

$$\pi_n(f) \;\colon\; \pi_n(X,x) \stackrel{\simeq}{\longrightarrow} \pi_n(Y,f(y)) \,.$$

Proposition

Every homotopy equivalence, def. , is a weak homotopy equivalence, def. .

In particular a deformation retraction, def. , is a weak homotopy equivalence.

Proof

First observe that for all $X\in$ Top the inclusion maps

$$X \overset{(id,\delta_0)}{\longrightarrow} X \times I$$

into the standard cylinder object, def. , are weak homotopy equivalences: by postcomposition with the contracting homotopy of the interval from example all homotopy groups of $X \times I$ have representatives that factor through this inclusion.

Then given a general homotopy equivalence, apply the homotopy groups functor to the corresponding homotopy diagrams (where for the moment we notationally suppress the choice of basepoint for readability) to get two commuting diagrams

$$\array{ \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(f)\circ \pi_\bullet(g)}} \\ \pi_\bullet(X \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(X) } \;\;\;\;\;\;\; \,, \;\;\;\;\;\;\; \array{ \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(g)\circ \pi_\bullet(f)}} \\ \pi_\bullet(Y \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(Y) } \,.$$

By the previous observation, the vertical morphisms here are isomorphisms, and hence these diagrams exhibit $\pi_\bullet(f)$ as the inverse of $\pi_\bullet(g)$, hence both as isomorphisms.

Remark

The converse of prop. is not true generally: not every weak homotopy equivalence between topological spaces is a homotopy equivalence. (For an example with full details spelled out see for instance Fritsch, Piccinini: “Cellular Structures in Topology”, p. 289-290).

However, as we will discuss below, it turns out that

1. every weak homotopy equivalence between CW-complexes is a homotopy equivalence (Whitehead's theorem, cor. );

2. every topological space is connected by a weak homotopy equivalence to a CW-complex (CW approximation, remark ).

Example

For $X\in Top$, the projection $X\times I \longrightarrow X$ from the cylinder object of $X$, def. , is a weak homotopy equivalence, def. . This means that the factorization

$$\nabla_X \;\colon\; X \sqcup X \stackrel{}{\hookrightarrow} X\times I \stackrel{\simeq}{\longrightarrow} X$$

of the codiagonal $\nabla_X$ in def. , which in general is far from being a monomorphism, may be thought of as factoring it through a monomorphism after replacing $X$, up to weak homotopy equivalence, by $X\times I$.

In fact, further below (prop. ) we see that $X \sqcup X \to X \times I$ has better properties than the generic monomorphism has, in particular better homotopy invariant properties: it has the left lifting property against all Serre fibrations $E \stackrel{p}{\longrightarrow} B$ that are also weak homotopy equivalences.

Of course the concept of left homotopy in def. is accompanied by a concept of right homotopy. This we turn to now.

Definition

(path space)

For $X$ a topological space, its standard topological path space object is the topological path space, hence the mapping space $X^I$, prop. , out of the standard interval $I$ of def. .

Example

The endpoint inclusion into the standard interval, def. , makes the path space $X^I$ of def. factor the diagonal on $X$ through the inclusion of constant paths and the endpoint evaluation of paths:

$$\Delta_X \;\colon\; X \stackrel{X^{I \to \ast}}{\longrightarrow} X^I \stackrel{X^{\ast \sqcup \ast \to I}}{\longrightarrow} X \times X \,.$$

This is the formal dual to example . As in that example, below we will see (prop. ) that this factorization has good properties, in that

1. $X^{I \to \ast}$ is a weak homotopy equivalence;

2. $X^{\ast \sqcup \ast \to I}$ is a Serre fibration.

So while in general the diagonal $\Delta_X$ is far from being an epimorphism or even just a Serre fibration, the factorization through the path space object may be thought of as replacing $X$, up to weak homotopy equivalence, by its path space, such as to turn its diagonal into a Serre fibration after all.

Definition

(right homotopy)

For $f,g\colon X \longrightarrow Y$ two continuous functions between topological spaces $X,Y$, then a right homotopy $f \Rightarrow_R g$ is a continuous function

$$\eta \;\colon\; X \longrightarrow Y^I$$

into the path space object of $X$, def. , such that this fits into a commuting diagram of the form