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Introduction to Homotopy Theory


This pages gives a detailed introduction to classical homotopy theory, starting with the concept of homotopy in topological spaces and motivating from this the “abstract homotopy theory” in general model categories.

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For background on basic topology see at Introduction to Topology.

For application to homological algebra see at Introduction to Homological algebra.

For application to stable homotopy theory see at Introduction to Stable homotopy theory.


Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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While the field of algebraic topology clearly originates in topology, it is not actually interested in topological spaces regarded up to topological isomorphism, namely homeomorphism (“point-set topology”), but only in topological spaces regarded up to weak homotopy equivalence – hence it is interested only in the “weak homotopy types” of topological spaces. This is so notably because ordinary cohomology groups are invariants of the (weak) homotopy type of topological spaces but do not detect their homeomorphism class.

The category of topological spaces obtained by forcing weak homotopy equivalences to become isomorphisms is the “classical homotopy categoryHo(Top). This homotopy category however has forgotten a little too much information: homotopy theory really wants the weak homotopy equivalences not to become plain isomorphisms, but to become actual homotopy equivalences. The structure that reflects this is called a model category structure (short for “category of models for homotopy types”). For classical homotopy theory this is accordingly called the classical model structure on topological spaces. This we review here.

Topological homotopy theory

This section 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 1 and lemma 5) that serve to establish the abstract homotopy theory of topological spaces further below.

Literature (Hirschhorn 15)

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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.

Generally, recall:

Definition

A diagram in a category 𝒞\mathcal{C} is a small category II and a functor

X :I𝒞 X_\bullet \;\colon\; I \longrightarrow \mathcal{C}
(iϕj)(X iX(ϕ)X j). (i \overset{\phi}{\longrightarrow} j) \; \mapsto ( X_i \overset{X(\phi)}{\longrightarrow} X_j) \,.

A cone over this diagram is an object QQ equipped with morphisms p i:QX ip_i \colon Q \longrightarrow X_i for all iIi \in I, such that all these triangles commute:

Q p i p j X i X(ϕ) X j. \array{ && Q \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ X_i && \underset{X(\phi)}{\longrightarrow} && X_j } \,.

Dually, a co-cone under the diagram is QQ equipped with morphisms q i:X iQq_i \colon X_i \longrightarrow Q such that all these triangles commute

X i X(ϕ) X j q i q j Q. \array{ X_i && \overset{X(\phi)}{\longrightarrow} && X_j \\ & {}_{\mathllap{q_i}}\searrow && \swarrow_{\mathrlap{q_j}} \\ && Q } \,.

A limit over the diagram is a universal cone, denoted lim iIX i\underset{\longleftarrow}{\lim}_{i \in I} X_i, that is: a cone such that every other cone uniquely factors through it Qlim iIX iQ \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} X_i, making all the resulting triangles commute.

Dually, a colimit over the diagram is a universal co-cone, denoted lim iIX i\underset{\longrightarrow}{\lim}_{i \in I} X_i.

We now discuss limits and colimits in 𝒞=\mathcal{C}= Top. The key for understanding these is the fact that there are initial and final topologies:

Definition

Let {X i=(S i,τ i)Top} iI\{X_i = (S_i,\tau_i) \in Top\}_{i \in I} be a set of topological spaces, and let SSetS \in Set be a bare set. Then

  1. For {Sf iS i} iI\{S \stackrel{f_i}{\to} S_i \}_{i \in I} a set of functions out of SS, the initial topology τ initial({f i} iI)\tau_{initial}(\{f_i\}_{i \in I}) is the topology on SS with the minimum collection of open subsets such that all f i:(S,τ initial({f i} iI))X if_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i are continuous.

  2. For {S if iS} iI\{S_i \stackrel{f_i}{\to} S\}_{i \in I} a set of functions into SS, the final topology τ final({f i} iI)\tau_{final}(\{f_i\}_{i \in I}) is the topology on SS with the maximum collection of open subsets such that all f i:X i(S,τ final({f i} iI))f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I})) are continuous.

Example

For XX a single topological space, and ι S:SU(X)\iota_S \colon S \hookrightarrow U(X) a subset of its underlying set, then the initial topology τ intial(ι S)\tau_{intial}(\iota_S), def. 2, is the subspace topology, making

ι S:(S,τ initial(ι S))X \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X

a topological subspace inclusion.

Example

Conversely, for p S:U(X)Sp_S \colon U(X) \longrightarrow S an epimorphism, then the final topology τ final(p S)\tau_{final}(p_S) on SS is the quotient topology.

Proposition

Let II be a small category and let X :ITopX_\bullet \colon I \longrightarrow Top be an II-diagram in Top (a functor from II to TopTop), with components denoted X i=(S i,τ i)X_i = (S_i, \tau_i), where S iSetS_i \in Set and τ i\tau_i a topology on S iS_i. Then:

  1. The limit of X 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. 2, for the functions p ip_i which are the limiting cone components:

    lim iIS i p i p j S i S j. \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.

    Hence

    lim iIX i(lim iIS i,τ initial({p i} iI)) \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 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. 2 for the component maps ι i\iota_i of the colimiting cocone

    S i S j ι i ι j lim iIS i. \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.

    Hence

    lim iIX i(lim iIS i,τ final({ι i} iI)) \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 (lim iIS i,τ initial({p i} iI))\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) (def. 1) is immediate: for

(S,τ) f i f j X i X j \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 Slim iIS iS \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 TopTop is the point *\ast with its unique topology.

Example

For {X i} iI\{X_i\}_{i \in I} a set of topological spaces, their coproduct iIX iTop\underset{i \in I}{\sqcup} X_i \in Top is their disjoint union.

In particular:

Example

For SSetS \in Set, the SS-indexed coproduct of the point, sS*\underset{s \in S}{\coprod}\ast is the set SS itself equipped with the final topology, hence is the discrete topological space on SS.

Example

For {X i} iI\{X_i\}_{i \in I} a set of topological spaces, their product iIX iTop\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 SS is a finite set, such as for binary product spaces X×YX \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:XYf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the equalizer of the underlying functions of sets

eq(f,g)S XgfS Y eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y

(hence the largets subset of S XS_X on which both functions coincide) and equipped with the subspace topology, example 1.

Example

The coequalizer of two continuous functions f,g:XYf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the coequalizer of the underlying functions of sets

S XgfS Ycoeq(f,g) 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)g(x)f(x) \sim g(x) for all xXx \in X) and equipped with the quotient topology, example 2.

Example

For

A g Y f X \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

A g Y f g *f X X AY.. \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,.

(Here g *fg_\ast f is also called the pushout of ff, or the cobase change of ff along gg.)

This is equivalently the coequalizer of the two morphisms from AA to the coproduct of XX with YY (example 4):

AXYX AY. A \stackrel{\longrightarrow}{\longrightarrow} X \sqcup Y \longrightarrow X \sqcup_A Y \,.

If gg is an inclusion, one also writes X fYX \cup_f Y and calls this the attaching space.

By example 8 the pushout/attaching space is the quotient topological space

X AY(XY)/ X \sqcup_A Y \simeq (X\sqcup Y)/\sim

of the disjoint union of XX and YY subject to the equivalence relation which identifies a point in XX with a point in YY if they have the same pre-image in AA.

(graphics from Aguilar-Gitler-Prieto 02)

Notice that the defining universal property of this colimit means that completing the span

A Y X \array{ A &\longrightarrow& Y \\ \downarrow \\ X }

to a commuting square

A Y X Z \array{ A &\longrightarrow& Y \\ \downarrow && \downarrow \\ X &\longrightarrow& Z }

is equivalent to finding a morphism

XAYZ. X \underset{A}{\sqcup} Y \longrightarrow Z \,.
Example

For AXA\hookrightarrow X a topological subspace inclusion, example 1, then the pushout

A X (po) * X/A \array{ A &\hookrightarrow& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A }

is the quotient space or cofiber, denoted X/AX/A.

Example

An important special case of example 9:

For nn \in \mathbb{N} write

  • D n{x n||x|1} nD^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 n1=D n{x n||x|=1} nS^{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=S^{-1} = \emptyset and that S 0=**S^0 = \ast \sqcup \ast.

Let

i n:S n1D n 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 n\mathbb{R}^n).

Then the colimit in Top under the diagram

D ni nS n1i nD n, D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \,,

i.e. the pushout of i ni_n along itself, is the n-sphere S nS^n:

S n1 i n D n i n (po) D n 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 SS equipped with a relation \leq such that for all elements a,b,cSa,b,c \in S

1) (reflexivity) aaa \leq a;

2) (transitivity) if aba \leq b and bcb \leq c then aca \leq c;

3) (antisymmetry) if aba\leq b and ba\b \leq a then a=ba = b.

This we may and will equivalently think of as a category with objects the elements of SS and a unique morphism aba \to b precisely if aba\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 a\bot \leq a for all a. A top element \top is one for wich aa \leq \top.

A partial order is a total order if in addition

4) (totality) either aba\leq b or bab \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 nn \in \mathbb{N}, corresponding to the well-orders {012n1}\{0 \leq 1 \leq 2 \cdots \leq n-1\}. Here (n+1)(n+1) is the successor of nn. The first non-empty limit ordinal is ω=[(,)]\omega = [(\mathbb{N}, \leq)].

Definition

Let 𝒞\mathcal{C} be a category, and let IMor(𝒞)I \subset Mor(\mathcal{C}) be a class of its morphisms.

For α\alpha an ordinal (regarded as a category), an α\alpha-indexed transfinite sequence of elements in II is a diagram

X :α𝒞 X_\bullet \;\colon\; \alpha \longrightarrow \mathcal{C}

such that

  1. X X_\bullet takes all successor morphisms ββ+1\beta \stackrel{\leq}{\to} \beta + 1 in α\alpha to elements in II

    X β,β+1I X_{\beta,\beta + 1} \in I
  2. X X_\bullet is continuous in that for every nonzero limit ordinal β<α\beta \lt \alpha, X X_\bullet restricted to the full-subdiagram {γ|γβ}\{\gamma \;|\; \gamma \leq \beta\} is a colimiting cocone in 𝒞\mathcal{C} for X X_\bullet restricted to {γ|γ<β}\{\gamma \;|\; \gamma \lt \beta\}.

The corresponding transfinite composition is the induced morphism

X 0X αlimX X_0 \longrightarrow X_\alpha \coloneqq \underset{\longrightarrow}{\lim}X_\bullet

into the colimit of the diagram, schematically:

X 0 X 0,1 X 1 X 1,2 X 2 X α. \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 XX a topological space and YY a locally compact topological space (in that for every point, every neighbourhood contains a compact neighbourhood), the mapping space

X YTop X^Y \in Top

is the topological space

Accordingly this is called the compact-open topology on the set of functions.

The construction extends to a functor

() ():Top lc op×TopTop. (-)^{(-)} \;\colon\; Top_{lc}^{op} \times Top \longrightarrow Top \,.
Proposition

For XX a topological space and YY a locally compact topological space (in that for each point, each open neighbourhood contains a compact neighbourhood), the topological mapping space X YX^Y from def. 5 is an exponential object, i.e. the functor () Y(-)^Y is right adjoint to the product functor Y×()Y \times (-): there is a natural bijection

Hom Top(Z×Y,X)Hom Top(Z,X Y) Hom_{Top}(Z \times Y, X) \simeq Hom_{Top}(Z, X^Y)

between continuous functions out of any product topological space of YY with any ZTopZ \in Top and continuous functions from ZZ 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. 2 it is often assumed that YY 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 2 fails in general if YY 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 cgTopTop_{cg} \hookrightarrow Top (def. 46) 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[0,1] 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

**(δ 0,δ 1)I!* \ast \sqcup \ast \stackrel{(\delta_0,\delta_1)}{\longrightarrow} I \stackrel{\exists !}{\longrightarrow} \ast

this is the standard interval object in Top.

For XTopX \in Top, the product topological space X×IX\times I, example 6, is called the standard cylinder object over XX. The endpoint inclusions of the interval make it factor the codiagonal on XX

X:XX((id,δ 0),(id,δ 1))X×IX. \nabla_X \;\colon\; X \sqcup X \stackrel{((id,\delta_0),(id,\delta_1))}{\longrightarrow} X \times I \longrightarrow X \,.
Definition

For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a left homotopy

η:f Lg \eta \colon f \,\Rightarrow_L\, g

is a continuous function

η:X×IY \eta \;\colon\; X \times I \longrightarrow Y

out of the standard cylinder object over XX, def. 6, such that this fits into a commuting diagram of the form

X (id,δ 0) f X×I η Y (id,δ 1) g X. \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 XX be a topological space and let x,yXx,y \in X be two of its points, regarded as functions x,y:*Xx,y \colon \ast \longrightarrow X from the point to XX. Then a left homotopy, def. 7, between these two functions is a commuting diagram of the form

* δ 0 x I η Y δ 1 y *. \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 XX whose endpoints are xx and yy.

For instance:

Example

Let

const 0:I*δ 0I const_0 \;\colon\; I \longrightarrow \ast \overset{\delta_0}{\longrightarrow} I

be the continuous function from the standard interval I=[0,1]I = [0,1] to itself that is constant on the value 0. Then there is a left homotopy, def. 7, from the identity function

η:id Iconst 0 \eta \;\colon\; id_I \Rightarrow const_0

given by

η(x,t)x(1t). \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 XX a topological space, then its set π 0(X)\pi_0(X) of connected components, also called the 0-th homotopy set, is the set of left homotopy-equivalence classes (def. 7) of points x:*Xx \colon \ast \to X, hence the set of path-connected components of XX (example 13). By composition this extends to a functor

π 0:TopSet. \pi_0 \colon Top \longrightarrow Set \,.

For nn \in \mathbb{N}, n1n \geq 1 and for x:*Xx \colon \ast \to X any point, then the nnth homotopy group π n(X,x)\pi_n(X,x) of XX at xx is the group

  • whose underlying set is the set of left homotopy-equivalence classes of maps I nXI^n \longrightarrow X that take the boundary of I nI^n to xx and where the left homotopies η\eta are constrained to be constant on the boundary;

  • whose group product operation takes [α:I nX][\alpha \colon I^n \to X] and [β:I nX][\beta \colon I^n \to X] to [αβ][\alpha \cdot \beta] with

αβ:I nI nI n1I n(α,β)X, \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 nn-cube to the nn-cube with one side twice the unit length (e.g. (x 1,x 2,x 3,)(2x 1,x 2,x 3,)(x_1, x_2, x_3, \cdots) \mapsto (2 x_1, x_2, x_3, \cdots)).

By composition, this construction extends to a functor

π 1:Top */Grp 1 \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 ff as f *=π (f)f_\ast = \pi_\bullet(f).

Remark

At this point we don’t go further into the abstract reason why def. 8 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

A continuous function f:XYf \;\colon\; X \longrightarrow Y is called a homotopy equivalence if there exists a continuous function the other way around, g:YXg \;\colon\; Y \longrightarrow X, and left homotopies, def. 7, from the two composites to the identity:

η 1:fg Lid Y \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y

and

η 2:gf Lid X. \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,.

If here η 2\eta_2 is constant along II, ff is said to exhibit XX as a deformation retract of YY.

Example

For XX a topological space and X×IX \times I its standard cylinder object of def. 6, then the projection p:X×IXp \colon X \times I \longrightarrow X and the inclusion (id,δ 0):XX×I(id, \delta_0) \colon X \longrightarrow X\times I are homotopy equivalences, def. 9, and in fact are homotopy inverses to each other:

The composition

X(id,δ 0)X×IpX X \overset{(id,\delta_0)}{\longrightarrow} X\times I \overset{p}{\longrightarrow} X

is immediately the identity on XX (i.e. homotopic to the identity by a trivial homotopy), while the composite

X×IpX(id,δ 0)X×I X \times I \overset{p}{\longrightarrow} X \overset{(id, \delta_0)}{\longrightarrow} X\times I

is homotopic to the identity on X×IX \times I by a homotopy that is pointwise in XX that of example 14.

Definition

A continuous function f:XYf \colon X \longrightarrow Y is called a weak homotopy equivalence if its image under all the homotopy group functors of def. 8 is an isomorphism, hence if

π 0(f):π 0(X)π 0(X) \pi_0(f) \;\colon\; \pi_0(X) \stackrel{\simeq}{\longrightarrow} \pi_0(X)

and for all xXx \in X and all n1n \geq 1

π n(f):π n(X,x)π n(Y,f(y)). \pi_n(f) \;\colon\; \pi_n(X,x) \stackrel{\simeq}{\longrightarrow} \pi_n(Y,f(y)) \,.
Proposition

Every homotopy equivalence, def. 9, is a weak homotopy equivalence, def. 10.

In particular a deformation retraction, def. 9, is a weak homotopy equivalence.

Proof

First observe that for all XX\in Top the inclusion maps

X(id,δ 0)X×I X \overset{(id,\delta_0)}{\longrightarrow} X \times I

into the standard cylinder object, def. 6, are weak homotopy equivalences: by postcomposition with the contracting homotopy of the interval from example 14 all homotopy groups of X×IX \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

π (X) π (id,δ 0) π (f)π (g) π (X×I) π (η) π (Y) π (id,δ 1) π (id) π (X),π (Y) π (id,δ 0) π (g)π (f) π (Y×I) π (η) π (X) π (id,δ 1) π (id) π (Y). \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 π (f)\pi_\bullet(f) as the inverse of π (g)\pi_\bullet(g), hence both as isomorphisms.

Remark

The converse of prop. 3 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. 3);

  2. every topological space is connected by a weak homotopy equivalence to a CW-complex (CW approximation, remark 16).

Example

For XTopX\in Top, the projection X×IXX\times I \longrightarrow X from the cylinder object of XX, def. 6, is a weak homotopy equivalence, def. 10. This means that the factorization

X:XXX×IX \nabla_X \;\colon\; X \sqcup X \stackrel{}{\hookrightarrow} X\times I \stackrel{\simeq}{\longrightarrow} X

of the codiagonal X\nabla_X in def. 6, which in general is far from being a monomorphism, may be thought of as factoring it through a monomorphism after replacing XX, up to weak homotopy equivalence, by X×IX\times I.

In fact, further below (prop. 14) we see that XXX×IX \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 EpB E \stackrel{p}{\longrightarrow} B that are also weak homotopy equivalences.

Of course the concept of left homotopy in def. 7 is accompanied by a concept of right homotopy. This we turn to now.

Definition

(path space)

For XX a topological space, its standard topological path space object is the topological path space, hence the mapping space X IX^I, prop. 2, out of the standard interval II of def. 6.

Example

The endpoint inclusion into the standard interval, def. 6, makes the path space X IX^I of def. 11 factor the diagonal on XX through the inclusion of constant paths and the endpoint evaluation of paths:

Δ X:XX I*X IX **IX×X. \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 6. As in that example, below we will see (prop. 22) that this factorization has good properties, in that

  1. X I*X^{I \to \ast} is a weak homotopy equivalence;

  2. X **IX^{\ast \sqcup \ast \to I} is a Serre fibration.

So while in general the diagonal Δ X\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 XX, up to weak homotopy equivalence, by its path space, such as to turn its diagonal into a Serre fibration after all.

Definition

For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a right homotopy f Rgf \Rightarrow_R g is a continuous function

η:XY I \eta \;\colon\; X \longrightarrow Y^I

into the path space object of XX, def. 11, such that this fits into a commuting diagram of the form

Y f X δ 0 X η Y I g Y δ 1 Y. \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{X^{\delta_0}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^I \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{\delta_1}}} \\ && Y } \,.

Cell complexes

We consider topological spaces that are built consecutively by attaching basic cells.

Definition

Write

I Top{S n1ι nD n} nMor(Top) I_{Top} \coloneqq \left\{ S^{n-1} \stackrel{\iota_n}{\hookrightarrow} D^{n} \right\}_{n \in \mathbb{N}} \; \subset Mor(Top)

for the set of canonical boundary inclusion maps of the standard n-disks, example 11. This going to be called the set of standard topological generating cofibrations.

Definition

For XTopX \in Top and for nn \in \mathbb{N}, an nn-cell attachment to XX is the pushout (“attaching space”, example 9) of a generating cofibration, def. 13

S n1 ϕ X ι n (po) D n XS n1D n =X ϕD n \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \underset{S^{n-1}}{\sqcup} D^n & = X \cup_\phi D^n }

along some continuous function ϕ\phi.

A continuous function f:XYf \colon X \longrightarrow Y is called a topological relative cell complex if it is exhibited by a (possibly infinite) sequence of cell attachments to XX, in that it is a transfinite composition (def. 4) of pushouts (example 9)

iS n i1 X k iι n i (po) iD n i X k+1 \array{ \underset{i}{\coprod} S^{n_i - 1} &\longrightarrow& X_{k} \\ {}^{\mathllap{\underset{i}{\coprod}\iota_{n_i}}}\downarrow &(po)& \downarrow \\ \underset{i}{\coprod} D^{n_i} &\longrightarrow& X_{k+1} }

of coproducts (example 4) of generating cofibrations (def. 13).

A topological space XX is a cell complex if X\emptyset \longrightarrow X is a relative cell complex.

A relative cell complex is called a finite relative cell complex if it is obtained from a finite number of cell attachments.

A (relative) cell complex is called a (relative) CW-complex if the above transfinite composition is countable

X=X 0 X 1 X 2 f Y=limX \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet }

and if X kX_k is obtained from X k1X_{k-1} by attaching cells precisely only of dimension kk.

Remark

Strictly speaking a relative cell complex, def. 14, is a function f:XYf\colon X \to Y, together with its cell structure, hence together with the information of the pushout diagrams and the transfinite composition of the pushout maps that exhibit it.

In many applications, however, all that matters is that there is some (relative) cell decomosition, and then one tends to speak loosely and mean by a (relative) cell complex only a (relative) topological space that admits some cell decomposition.

The following lemma 1, together with lemma 5 below are the only two statements of the entire development here that involve the concrete particular nature of topological spaces (“point-set topology”), everything beyond that is general abstract homotopy theory.

Lemma

Assuming the axiom of choice and the law of excluded middle, every compact subspace of a topological cell complex, def. 14, intersects the interior of a finite number of cells.

(e.g. Hirschhorn 15, section 3.1)

Proof

So let YY be a topological cell complex and CYC \hookrightarrow Y a compact subspace. Define a subset

PY P \subset Y

by choosing one point in the interior of the intersection with CC of each cell of YY that intersects CC.

It is now sufficient to show that PP has no accumulation point. Because, by the compactness of XX, every non-finite subset of CC does have an accumulation point, and hence the lack of such shows that PP is a finite set and hence that CC intersects the interior of finitely many cells of YY.

To that end, let cCc\in C be any point. If cc is a 0-cell in YY, write U c{c}U_c \coloneqq \{c\}. Otherwise write e ce_c for the unique cell of YY that contains cc in its interior. By construction, there is exactly one point of PP in the interior of e ce_c. Hence there is an open neighbourhood cU ce cc \in U_c \subset e_c containing no further points of PP beyond possibly cc itself, if cc happens to be that single point of PP in e ce_c.

It is now sufficient to show that U cU_c may be enlarged to an open subset U˜ c\tilde U_c of YY containing no point of PP, except for possibly cc itself, for that means that cc is not an accumulation point of PP.

To that end, let α c\alpha_c be the ordinal that labels the stage Y α cY_{\alpha_c} of the transfinite composition in the cell complex-presentation of YY at which the cell e ce_c above appears. Let γ\gamma be the ordinal of the full cell complex. Then define the set

T{(β,U)|α cβγ,UopenY β,UY α=U c,UP{,{c}}}, T \coloneqq \left\{ \; (\beta, U) \;|\; \alpha_c \leq \beta \leq \gamma \;\,,\; U \underset{open}{\subset} Y_\beta \;\,,\; U \cap Y_\alpha = U_c \;\,,\; U \cap P \in \{ \emptyset, \{c\} \} \; \right\} \,,

and regard this as a partially ordered set by declaring a partial ordering via

(β 1,U 1)<(β 2,U 2)β 1<β 2,U 2Y β 1=U 1. (\beta_1, U_1) \lt (\beta_2, U_2) \;\;\;\; \Leftrightarrow \;\;\;\; \beta_1 \lt \beta_2 \;\,,\; U_2 \cap Y_{\beta_1} = U_1 \,.

This is set up such that every element (β,U)(\beta, U) of TT with β\beta the maximum value β=γ\beta = \gamma is an extension U˜ c\tilde U_c that we are after.

Observe then that for (β s,U s) sS(\beta_s, U_s)_{s\in S} a chain in (T,<)(T,\lt) (a subset on which the relation <\lt restricts to a total order), it has an upper bound in TT given by the union ( sβ s, sU s)({\cup}_s \beta_s ,\cup_s U_s). Therefore Zorn's lemma applies, saying that (T,<)(T,\lt) contains a maximal element (β max,U max)(\beta_{max}, U_{max}).

Hence it is now sufficient to show that β max=γ\beta_{max} = \gamma. We argue this by showing that assuming β max<γ\beta_{\max}\lt \gamma leads to a contradiction.

So assume β max<γ\beta_{max}\lt \gamma. Then to construct an element of TT that is larger than (β max,U max)(\beta_{max},U_{max}), consider for each cell dd at stage Y β max+1Y_{\beta_{max}+1} its attaching map h d:S n1Y β maxh_d \colon S^{n-1} \to Y_{\beta_{max}} and the corresponding preimage open set h d 1(U max)S n1h_d^{-1}(U_{max})\subset S^{n-1}. Enlarging all these preimages to open subsets of D nD^n (such that their image back in X β max+1X_{\beta_{max}+1} does not contain cc), then (β max,U max)<(β max+1, dU d)(\beta_{max}, U_{max}) \lt (\beta_{max}+1, \cup_d U_d ). This is a contradiction. Hence β max=γ\beta_{max} = \gamma, and we are done.

It is immediate and useful to generalize the concept of topological cell complexes as follows.

Definition

For 𝒞\mathcal{C} any category and for KMor(𝒞)K \subset Mor(\mathcal{C}) any sub-class of its morphisms, a relative KK-cell complexes is a morphism in 𝒞\mathcal{C} which is a transfinite composition (def. 4) of pushouts of coproducts of morphsims in KK.

Definition

Write

J Top{D n(id,δ 0)D n×I} nMor(Top) J_{Top} \coloneqq \left\{ D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I \right\}_{n \in \mathbb{N}} \; \subset Mor(Top)

for the set of inclusions of the topological n-disks, def. 13, into their cylinder objects, def. 6, along (for definiteness) the left endpoint inclusion.

These inclusions are similar to the standard topological generating cofibrations I TopI_{Top} of def. 13, but in contrast to these they are “acyclic” (meaning: trivial on homotopy classes of maps from “cycles” given by n-spheres) in that they are weak homotopy equivalences (by prop. 3).

Accordingly, J TopJ_{Top} is to be called the set of standard topological generating acyclic cofibrations.

Lemma

For XX a CW-complex (def. 14), then its inclusion X(id,δ 0)X×IX \overset{(id, \delta_0)}{\longrightarrow} X\times I into its standard cylinder (def. 6) is a J TopJ_{Top}-relative cell complex (def. 15, def. 16).

Proof

First erect a cylinder over all 0-cells

xX 0D 0 X (po) xX 0D 1 Y 1. \array{ \underset{x \in X_0}{\coprod} D^0 &\longrightarrow& X \\ \downarrow &(po)& \downarrow \\ \underset{x\in X_0}{\coprod} D^1 &\longrightarrow& Y_1 } \,.

Assume then that the cylinder over all nn-cells of XX has been erected using attachment from J TopJ_{Top}. Then the union of any (n+1)(n+1)-cell σ\sigma of XX with the cylinder over its boundary is homeomorphic to D n+1D^{n+1} and is like the cylinder over the cell “with end and interior removed”. Hence via attaching along D n+1D n+1×ID^{n+1} \to D^{n+1}\times I the cylinder over σ\sigma is erected.

Lemma

The maps D nD n×ID^n \hookrightarrow D^n \times I in def. 16 are finite relative cell complexes, def. 14. In other words, the elements of J TopJ_{Top} are I TopI_{Top}-relative cell complexes.

Proof

There is a homeomorphism

D n = D n (id,δ 0) D n×I D n+1 \array{ D^n & = & D^n \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow \\ D^n \times I &\simeq& D^{n+1} }

such that the map on the right is the inclusion of one hemisphere into the boundary n-sphere of D n+1D^{n+1}. This inclusion is the result of attaching two cells:

S n1 ι n D n ι n (po) D n S n = S n id S n ι n+1 (po) D n+1 id D n+1. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^{n} \\ && \downarrow^{=} \\ S^n &\overset{id}{\longrightarrow}& S^n \\ {}^{\mathllap{\iota_{n+1}}}\downarrow &(po)& \downarrow \\ D^{n+1} &\underset{id}{\longrightarrow}& D^{n+1} } \,.

here the top pushout is the one from example 11.

Lemma

Every J TopJ_{Top}-relative cell complex (def. 16, def. 15) is a weak homotopy equivalence, def. 10.

Proof

Let XX^=lim βαX βX \longrightarrow \hat X = \underset{\longleftarrow}{\lim}_{\beta \leq \alpha} X_\beta be a J TopJ_{Top}-relative cell complex.

First observe that with the elements D nD n×ID^n \hookrightarrow D^n \times I of J TopJ_{Top} being homotopy equivalences for all nn \in \mathbb{N} (by example 15), each of the stages X βX β+1X_{\beta} \longrightarrow X_{\beta + 1} in the relative cell complex is also a homotopy equivalence. We make this fully explicit:

By definition, such a stage is a pushout of the form

iID n i X β iI(id,δ 0) (po) iID n i×I X β+1. \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} } \,.

Then the fact that the projections p n i:D n i×ID n ip_{n_i} \colon D^{n_i} \times I \to D^{n_i} are strict left inverses to the inclusions (id,δ 0)(id, \delta_0) gives a commuting square of the form

iID n i X β iI(id,δ 0) id iID n i×I iIp n i iID n i X β \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow && \downarrow^{\mathrlap{id}} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I \\ {}^{\mathllap{\underset{i \in I}{\sqcup} p_{n_i} }}\downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta }

and so the universal property of the colimit (pushout) X β+1X_{\beta + 1} gives a factorization of the identity morphism on the right through X β+1X_{\beta + 1}

iID n i X β iI(id,δ 0) iID n i×I X β+1 iIp n i iID n i X β \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow && \downarrow^{} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} \\ {}^{\mathllap{\underset{i \in I}{\sqcup} p_{n_i} }}\downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta }

which exhibits X β+1X βX_{\beta + 1} \to X_\beta as a strict left inverse to X βX β+1X_{\beta} \to X_{\beta + 1}. Hence it is now sufficient to show that this is also a homotopy right inverse.

To that end, let

η n i:D n i×ID n i×I \eta_{n_i} \colon D^{n_i}\times I \longrightarrow D^{n_i} \times I

be the left homotopy that exhibits p n ip_{n_i} as a homotopy right inverse to p n ip_{n_i} by example 15. For each t[0,1]t \in [0,1] consider the commuting square

iID n i X β iID n i×I X β+1 η n i(,t) id iID n i×I X β+1. \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ \downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} \times I && X_{\beta + 1} \\ {}^{\mathllap{\eta_{n_i}(-,t)}}\downarrow && \downarrow^{\mathrlap{id}} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} } \,.

Regarded as a cocone under the span in the top left, the universal property of the colimit (pushout) X β+1X_{\beta + 1} gives a continuous function

η(,t):X β+1X β+1 \eta(-,t) \;\colon\; X_{\beta + 1} \longrightarrow X_{\beta + 1}

for each t[0,1]t \in [0,1]. For t=0t = 0 this construction reduces to the provious one in that η(,0):X β+1X βX β+1\eta(-,0) \colon X_{\beta +1 } \to X_{\beta} \to X_{\beta + 1} is the composite which we need to homotope to the identity; while η(,1)\eta(-,1) is the identity. Since η(,t)\eta(-,t) is clearly also continuous in tt it constitutes a continuous function

η:X β+1×IX β+1 \eta \;\colon\; X_{\beta + 1}\times I \longrightarrow X_{\beta + 1}

which exhibits the required left homotopy.

So far this shows that each stage X βX β+1X_{\beta} \to X_{\beta+1} in the transfinite composition defining X^\hat X is a homotopy equivalence, hence, by prop. 3, a weak homotopy equivalence.

This means that all morphisms in the following diagram (notationally suppressing basepoints and showing only the finite stages)

π n(X) π n(X 1) π n(X 2) π n(X 3) lim απ n(X α) \array{ \pi_n(X) &\overset{\simeq}{\longrightarrow}& \pi_n(X_1) &\overset{\simeq}{\longrightarrow}& \pi_n(X_2) &\overset{\simeq}{\longrightarrow}& \pi_n(X_3) &\overset{\simeq}{\longrightarrow}& \cdots \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\simeq}} & \swarrow_{\mathrlap{\simeq}} & \cdots \\ && \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) }

are isomorphisms.

Moreover, lemma 1 gives that every representative and every null homotopy of elements in π n(X^)\pi_n(\hat X) already exists at some finite stage X kX_k. This means that also the universally induced morphism

lim απ n(X α)π n(X^) \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) \overset{\simeq}{\longrightarrow} \pi_n(\hat X)

is an isomorphism. Hence the composite π n(X)π n(X^)\pi_n(X) \overset{\simeq}{\longrightarrow} \pi_n(\hat X) is an isomorphism.

Fibrations

Given a relative CC-cell complex ι:XY\iota \colon X \to Y, def. 15, it is typically interesting to study the extension problem along ff, i.e. to ask which topological spaces EE are such that every continuous function f:XEf\colon X \longrightarrow E has an extension f˜\tilde f along ι\iota

X f E ι f˜ Y. \array{ X &\stackrel{f}{\longrightarrow}& E \\ {}^{\mathllap{\iota}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}} \\ Y } \,.

If such extensions exists, it means that EE is sufficiently “spread out” with respect to the maps in CC. More generally one considers this extension problem fiberwise, i.e. with both EE and YY (hence also XX) equipped with a map to some base space BB:

Definition

Given a category 𝒞\mathcal{C} and a sub-class CMor(𝒞)C \subset Mor(\mathcal{C}) of its morphisms, then a morphism p:EBp \colon E \longrightarrow B in 𝒞\mathcal{C} is said to have the right lifting property against the morphisms in CC if every commuting diagram in 𝒞\mathcal{C} of the form

X E c p Y B, \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,,

with cCc \in C, has a lift hh, in that it may be completed to a commuting diagram of the form

X E c h p Y B. \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,.

We will also say that ff is a CC-injective morphism if it satisfies the right lifting property against CC.

Definition

A continuous function p:EBp \colon E \longrightarrow B is called a Serre fibration if it is a J TopJ_{Top}-injective morphism; i.e. if it has the right lifting property, def. 17, against all topological generating acylic cofibrations, def. 16; hence if for every commuting diagram of continuous functions of the form

D n E (id,δ 0) p D n×I B, \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,,

has a lift hh, in that it may be completed to a commuting diagram of the form

D n E (id,δ 0) h p D n×I B. \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,.
Remark

Def. 18 says, in view of the definition of left homotopy, that a Serre fibration pp is a map with the property that given a left homotopy, def. 7, between two functions into its codomain, and given a lift of one the two functions through pp, then also the homotopy between the two lifts. Therefore the condition on a Serre fibration is also called the homotopy lifting property for maps whose domain is an n-disk.

More generally one may ask functions pp to have such homotopy lifting property for functions with arbitrary domain. These are called Hurewicz fibrations.

Remark

The precise shape of D nD^n and D n×ID^n \times I in def. 18 turns out not to actually matter much for the nature of Serre fibrations. We will eventually find below (prop. 17) that what actually matters here is only that the inclusions D nD n×ID^n \hookrightarrow D^n \times I are relative cell complexes (lemma 3) and weak homotopy equivalences (prop. 3) and that all of these may be generated from them in a suitable way.

But for simple special cases this is readily seen directly, too. Notably we could replace the n-disks in def. 18 with any homeomorphic topological space. A choice important in the comparison to the classical model structure on simplicial sets (below) is to instead take the topological n-simplices Δ n\Delta^n. Hence a Serre fibration is equivalently characterized as having lifts in all diagrams of the form

Δ n E (id,δ 0) p Δ n×I B. \array{ \Delta^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ \Delta^n\times I &\longrightarrow& B } \,.

Other deformations of the nn-disks are useful in computations, too. For instance there is a homeomorphism from the nn-disk to its “cylinder with interior and end removed”, formally:

(D n×{0})(D n×I) D n D n×I D n×I \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\simeq& D^n \\ \downarrow && \downarrow \\ D^n \times I &\simeq& D^n\times I }

and hence ff is a Serre fibration equivalently also if it admits lifts in all diagrams of the form

(D n×{0})(D n×I) E (id,δ 0) p D n×I B. \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,.

The following is a general fact about closure of morphisms defined by lifting properties which we prove in generality below as prop. 6.

Proposition

A Serre fibration, def. 18 has the right lifting property against all retracts (see remark 11) of J TopJ_{Top}-relative cell complexes (def. 16, def. 14).

The following statement is foreshadowing the long exact sequences of homotopy groups (below) induced by any fiber sequence, the full version of which we come to below (example 44) after having developed more of the abstract homotopy theory.

Proposition

Let f:XYf\colon X \longrightarrow Y be a Serre fibration, def. 18, let y:*Yy \colon \ast \to Y be any point and write

F yιXfY F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y

for the fiber inclusion over that point. Then for every choice x:*Xx \colon \ast \to X of lift of the point yy through ff, the induced sequence of homotopy groups

π (F y,x)ι *π (X,x)f *π (Y) \pi_{\bullet}(F_y, x) \overset{\iota_\ast}{\longrightarrow} \pi_\bullet(X, x) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y)

is exact, in that the kernel of f *f_\ast is canonically identified with the image of ι *\iota_\ast:

ker(f *)im(ι *). ker(f_\ast) \simeq im(\iota_\ast) \,.
Proof

It is clear that the image of ι *\iota_\ast is in the kernel of f *f_\ast (every sphere in F yXF_y\hookrightarrow X becomes constant on yy, hence contractible, when sent forward to YY).

For the converse, let [α]π (X,x)[\alpha]\in \pi_{\bullet}(X,x) be represented by some α:S n1X\alpha \colon S^{n-1} \to X. Assume that [α][\alpha] is in the kernel of f *f_\ast. This means equivalently that α\alpha fits into a commuting diagram of the form

S n1 α X f D n κ Y, \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y } \,,

where κ\kappa is the contracting homotopy witnessing that f *[α]=0f_\ast[\alpha] = 0.

Now since xx is a lift of yy, there exists a left homotopy

η:κconst y \eta \;\colon\; \kappa \Rightarrow const_y

as follows:

S n1 α X ι n f D n κ Y (id,δ 1) id D n (id,δ 0) D n×I η Y * y Y \array{ && S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ && D^n &\overset{\kappa}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{(id,\delta_1)}} && \downarrow^{\mathrlap{id}} \\ D^n &\overset{(id,\delta_0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow \\ \ast && \overset{y}{\longrightarrow} && Y }

(for instance: regard D nD^n as embedded in n\mathbb{R}^n such that 0 n0 \in \mathbb{R}^n is identified with the basepoint on the boundary of D nD^n and set η(v,t)κ(tv)\eta(\vec v,t) \coloneqq \kappa(t \vec v)).

The pasting of the top two squares that have appeared this way is equivalent to the following commuting square

S n1 α X (id,δ 1) f S n1×I (ι n,id) D n×I η Y. \array{ S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{(id,\delta_1)}}\downarrow && && \downarrow^{\mathrlap{f}} \\ S^{n-1} \times I &\overset{(\iota_n, id)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y } \,.

Because ff is a Serre fibration and by lemma 2 and prop. 4, this has a lift

η˜:S n1×IX. \tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X \,.

Notice that η˜\tilde \eta is a basepoint preserving left homotopy from α=η˜| 1\alpha = \tilde \eta|_1 to some αη˜| 0\alpha' \coloneqq \tilde \eta|_0. Being homotopic, they represent the same element of π n1(X,x)\pi_{n-1}(X,x):

[α]=[α]. [\alpha'] = [\alpha] \,.

But the new representative α\alpha' has the special property that its image in YY is not just trivializable, but trivialized: combining η˜\tilde \eta with the previous diagram shows that it sits in the following commuting diagram

α: S n1 (id,δ 0) S n1×I η˜ X ι n (ι n,id) f D n (id,δ 0) D n×I η Y * y Y. \array{ \alpha' \colon & S^{n-1} &\overset{(id,\delta_0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X \\ & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}} \\ & D^n &\overset{(id,\delta_0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ & \downarrow && && \downarrow \\ & \ast && \overset{y}{\longrightarrow} && Y } \,.

The commutativity of the outer square says that f *αf_\ast \alpha' is constant, hence that α\alpha' is entirely contained in the fiber F yF_y. Said more abstractly, the universal property of fibers gives that α\alpha' factors through F yιXF_y\overset{\iota}{\hookrightarrow} X, hence that [α]=[α][\alpha'] = [\alpha] is in the image of ι *\iota_\ast.

The following lemma 5, together with lemma 1 above, are the only two statements of the entire development here that crucially involve the concrete particular nature of topological spaces (“point-set topology”), everything beyond that is general abstract homotopy theory.

Lemma

The continuous functions with the right lifting property, def. 17 against the set I Top={S n1D n}I_{Top} = \{S^{n-1}\hookrightarrow D^n\} of topological generating cofibrations, def. 13, are precisely those which are both weak homotopy equivalences, def. 10 as well as Serre fibrations, def. 18.

Proof

We break this up into three sub-statements:

A) I TopI_{Top}-injective morphisms are in particular weak homotopy equivalences

Let p:X^Xp \colon \hat X \to X have the right lifting property against I TopI_{Top}

S n1 X^ ι n p D n X \array{ S^{n-1} &\longrightarrow & \hat X \\ {}^{\mathllap{\iota_n}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n &\longrightarrow& X }

We check that the lifts in these diagrams exhibit π (f)\pi_\bullet(f) as being an isomorphism on all homotopy groups, def. 8:

For n=0n = 0 the existence of these lifts says that every point of XX is in the image of pp, hence that π 0(X^)π 0(X)\pi_0(\hat X) \to \pi_0(X) is surjective. Let then S 0=**X^S^0 = \ast \coprod \ast \longrightarrow \hat X be a map that hits two connected components, then the existence of the lift says that if they have the same image in π 0(X)\pi_0(X) then they were already the same connected component in X^\hat X. Hence π 0(X^)π 0(X)\pi_0(\hat X)\to \pi_0(X) is also injective and hence is a bijection.

Similarly, for n1n \geq 1, if S nX^S^n \to \hat X represents an element in π n(X^)\pi_n(\hat X) that becomes trivial in π n(X)\pi_n(X), then the existence of the lift says that it already represented the trivial element itself. Hence π n(X^)π n(X)\pi_n(\hat X) \to \pi_n(X) has trivial kernel and so is injective.

Finally, to see that π n(X^)π n(X)\pi_n(\hat X) \to \pi_n(X) is also surjective, hence bijective, observe that every elements in π n(X)\pi_n(X) is equivalently represented by a commuting diagram of the form

S n1 * X^ D n X = X \array{ S^{n-1} &\longrightarrow& \ast &\longrightarrow& \hat X \\ \downarrow && \downarrow && \downarrow \\ D^n &\longrightarrow& X &=& X }

and so here the lift gives a representative of a preimage in π n(X^)\pi_{n}(\hat X).

B) I TopI_{Top}-injective morphisms are in particular Serre fibrations

By an immediate closure property of lifting problems (we spell this out in generality as prop. 6, cor. 7 below) an I TopI_{Top}-injective morphism has the right lifting property against all relative cell complexes, and hence, by lemma 3, it is also a J TopJ_{Top}-injective morphism, hence a Serre fibration.

C) Acyclic Serre fibrations are in particular I TopI_{Top}-injective morphisms

(Hirschhorn 15, section 6).

Let f:XYf\colon X \to Y be a Serre fibration that induces isomorphisms on homotopy groups. In degree 0 this means that ff is an isomorphism on connected components, and this means that there is a lift in every commuting square of the form

S 1= X f D 0=* Y \array{ S^{-1} = \emptyset &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^0 = \ast &\longrightarrow& Y }

(this is π 0(f)\pi_0(f) being surjective) and in every commuting square of the form

S 0 X ι 0 f D 1=* Y \array{ S^0 &\longrightarrow& X \\ {}^{\mathllap{\iota_0}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^1 = \ast &\longrightarrow& Y }

(this is π 0(f)\pi_0(f) being injective). Hence we are reduced to showing that for n2n \geq 2 every diagram of the form

S n1 α X ι n f D n κ Y \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y }

has a lift.

To that end, pick a basepoint on S n1S^{n-1} and write xx and yy for its images in XX and YY, respectively

Then the diagram above expresses that f *[α]=0π n1(Y,y)f_\ast[\alpha] = 0 \in \pi_{n-1}(Y,y) and hence by assumption on ff it follows that [α]=0π n1(X,x)[\alpha] = 0 \in \pi_{n-1}(X,x), which in turn mean that there is κ\kappa' making the upper triangle of our lifting problem commute:

S n1 α X ι n κ D n. \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & \nearrow_{\mathrlap{\kappa'}} \\ D^n } \,.

It is now sufficient to show that any such κ\kappa' may be deformed to a ρ\rho' which keeps making this upper triangle commute but also makes the remaining lower triangle commute.

To that end, notice that by the commutativity of the original square, we already have at least this commuting square:

S n1 ι n D n ι n fκ D n κ Y. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& Y } \,.

This induces the universal map (κ,fκ)(\kappa,f \circ \kappa') from the pushout of its cospan in the top left, which is the n-sphere (see this example):

S n1 ι n D n ι n (po) fκ D n κ S n (κ,fκ) Y. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& S^n \\ && & \searrow^{(\kappa,f \circ \kappa')} \\ && && Y } \,.

This universal morphism represents an element of the nnth homotopy group:

[(κ,fκ)]π n(Y,y). [(\kappa,f \circ \kappa')] \in \pi_n(Y,y) \,.

By assumption that ff is a weak homotopy equivalence, there is a [ρ]π n(X,x)[\rho] \in \pi_{n}(X,x) with

f *[ρ]=[(κ,fκ)] f_\ast [\rho] = [(\kappa,f \circ \kappa')]

hence on representatives there is a lift up to homotopy

X ρ f S n (κ,fκ) Y. \array{ && X \\ &{}^{\mathllap{\rho}}\nearrow_{\mathrlap{\Downarrow}} & \downarrow^{\mathrlap{f}} \\ S^n &\underset{(\kappa,f\circ \kappa')}{\longrightarrow}& Y } \,.

Morever, we may always find ρ\rho of the form (ρ,κ)(\rho', \kappa') for some ρ:D nX\rho' \colon D^n \to X. (“Paste κ\kappa' to the reverse of ρ\rho.”)

Consider then the map

S n(fρ,κ)Y S^n \overset{(f\circ \rho', \kappa)}{\longrightarrow} Y

and observe that this represents the trivial class:

[(fρ,κ)] =[(fρ,fκ)]+[(fκ,κ)] =f *[(ρ,κ)]=[ρ]+[(fκ,κ)] =[(κ,fκ)]+[(fκ,κ)] =0. \begin{aligned} [(f \circ \rho', \kappa)] & = [(f\circ \rho', f\circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = f_\ast \underset{= [\rho]}{\underbrace{[(\rho',\kappa')]}} + [(f\circ \kappa', \kappa)] \\ & = [(\kappa,f \circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = 0 \end{aligned} \,.

This means equivalently that there is a homotopy

ϕ:fρκ \phi \; \colon \; f\circ \rho' \Rightarrow \kappa

fixing the boundary of the nn-disk.

Hence if we denote homotopy by double arrows, then we have now achieved the following situation

S n1 α X ι n ρ ϕ f D n Y \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & {}^{\rho'}\nearrow_{\Downarrow^{\phi}} & \downarrow^{\mathrlap{f}} \\ D^n &\longrightarrow& Y }

and it now suffices to show that ϕ\phi may be lifted to a homotopy of just ρ\rho', fixing the boundary, for then the resulting homotopic ρ\rho'' is the desired lift.

To that end, notice that the condition that ϕ:D n×IY\phi \colon D^n \times I \to Y fixes the boundary of the nn-disk means equivalently that it extends to a morphism

S n1S n1×ID n×I(fα,ϕ)Y S^{n-1} \underset{S^{n-1}\times I}{\sqcup} D^n \times I \overset{(f\circ \alpha,\phi)}{\longrightarrow} Y

out of the pushout that identifies in the cylinder over D nD^n all points lying over the boundary. Hence we are reduced to finding a lift in

D n ρ X f S n1S n1×ID n×I (fα,ϕ) Y. \array{ D^n &\overset{\rho'}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ S^{n-1}\underset{S^{n-1}\times I}{\sqcup} D^n \times I &\overset{(f\circ \alpha,\phi)}{\longrightarrow}& Y } \,.

But inspection of the left map reveals that it is homeomorphic again to D nD n×ID^n \to D^n \times I, and hence the lift does indeed exist.

Abstract homotopy theory

In the above we discussed three classes of continuous functions between topological spaces

  1. weak homotopy equivalences;

  2. relative cell complexes;

  3. Serre fibrations

and we saw first aspects of their interplay via lifting properties.

A fundamental insight due to (Quillen 67) is that in fact all constructions in homotopy theory are elegantly expressible via just the abstract interplay of these classes of morphisms. This was distilled in (Quillen 67) into a small set of axioms called a model category structure (because it serves to make all objects be models for homotopy types.)

This abstract homotopy theory is the royal road for handling any flavor of homotopy theory, in particular the stable homotopy theory that we are after in Part 1. Here we discuss the basic constructions and facts in abstract homotopy theory, then below we conclude section P1) by showing that the above system of classes of maps of topological spaces is indeed an example.

Literature (Dwyer-Spalinski 95)

\,

Definition

A category with weak equivalences is

  1. a category 𝒞\mathcal{C};

  2. a sub-class WMor(𝒞)W \subset Mor(\mathcal{C}) of its morphisms;

such that

  1. WW contains all the isomorphisms of 𝒞\mathcal{C};

  2. WW is closed under two-out-of-three: in every commuting diagram in 𝒞\mathcal{C} of the form

    Y X Z \array{ && Y \\ & \nearrow&& \searrow \\ X && \longrightarrow && Z }

    if two of the three morphisms are in WW, then so is the third.

Remark

It turns out that a category with weak equivalences, def. 19, already determines a homotopy theory: the one given given by universally forcing weak equivalences to become actual homotopy equivalences. This may be made precise and is called the simplicial localization of a category with weak equivalences (Dwyer-Kan 80a, Dwyer-Kan 80b, Dwyer-Kan 80c). However, without further auxiliary structure, these simplicial localizations are in general intractable. The further axioms of a model category serve the sole purpose of making the universal homotopy theory induced by a category with weak equivalences be tractable:

Definition

A model category is

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

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

such that

  1. the class WW makes 𝒞\mathcal{C} into a category with weak equivalences, def. 19;

  2. The pairs (WCof,Fib)(W \cap Cof\;,\; Fib) and (Cap,WFib)(Cap\;,\; W\cap Fib) are both weak factorization systems, def. 22.

One says:

The form of def. 20 is due to (Joyal, def. E.1.2). It implies various other conditions that (Quillen 67) demands explicitly, see prop. 6 and prop. 8 below.

We now dicuss the concept of weak factorization systems appearing in def. 20.

Factorization systems

Definition

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

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

then an extension of the morphism ff along the morphism pp is a completion to a commuting diagram of the form

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

Dually, given a diagram of the form

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

then a lift of ff through pp is a completion to a commuting diagram of the form

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

Combining these cases: given a commuting square

X 1 f 1 Y 1 p l p r X 2 f 1 Y 2 \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

X 1 f 1 Y 1 p l p r X 2 f 1 Y 2. \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 KMor(𝒞)K \subset Mor(\mathcal{C}), then

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

dually:

  • a morphism p lp_l is said to have the left lifting property against KK or to be a KK-projective morphism if in all square diagrams with p lp_l on the left and any p rKp_r \in K on the left a lift exists.
Definition

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

  1. Every morphism f:XYf \colon X\to Y of 𝒞\mathcal{C} may be factored as the composition of a morphism in ProjProj followed by one in InjInj

    f:XProjZInjY. f\;\colon\; X \overset{\in Proj}{\longrightarrow} Z \overset{\in Inj}{\longrightarrow} Y \,.
  2. The classes are closed under having the lifting property, def. 21, against each other:

    1. ProjProj is precisely the class of morphisms having the left lifting property against every morphisms in InjInj;

    2. InjInj is precisely the class of morphisms having the right lifting property against every morphisms in ProjProj.

Definition

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

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

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

Remark

In def. 23 we are using the following standard notation, see at simplex category and at nerve of a category:

Write [1]={01}[1] = \{0 \to 1\} and [2]={012}[2] = \{0 \to 1 \to 2\} for the ordinal numbers, regarded as posets and hence as categories. The arrow category Arr(𝒞)Arr(\mathcal{C}) is equivalently the functor category 𝒞 Δ[1]Funct(Δ[1],𝒞)\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C}), while 𝒞 Δ[2]Funct(Δ[2],𝒞)\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C}) has as objects pairs of composable morphisms in 𝒞\mathcal{C}. There are three injective functors δ i:[1][2]\delta_i \colon [1] \rightarrow [2], where δ i\delta_i omits the index ii in its image. By precomposition, this induces functors d i:𝒞 Δ[2]𝒞 Δ[1]d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}. Here

  • d 1d_1 sends a pair of composable morphisms to their composition;

  • d 2d_2 sends a pair of composable morphisms to the first morphisms;

  • d 0d_0 sends a pair of composable morphisms to the second morphisms.

Definition

A weak factorization system, def. 22, is called a functorial weak factorization system if the factorization of morphisms may be chosen to be a functorial factorization factfact, def. 23, i.e. such that d 2factd_2 \circ fact lands in ProjProj and d 0factd_0\circ fact in InjInj.

Remark

Not all weak factorization systems are functorial, def. 24, although most (including those produced by the small object argument (prop. 9 below), with due care) are.

Proposition

Let 𝒞\mathcal{C} be a category and let KMor(𝒞)K\subset Mor(\mathcal{C}) be a class of morphisms. Write KProjK Proj and KInjK Inj, respectively, for the sub-classes of KK-projective morphisms and of KK-injective morphisms, def. 21. Then:

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

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

    KProjK Proj is also closed under transfinite composition.

  3. Both classes are closed under forming retracts in the arrow category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} (see remark 11).

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

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

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

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

Proof

We go through each item in turn.

containing isomorphisms

Given a commuting square

A f X Iso i p B g Y \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 fi 1{}^{{f \circ i^{-1}}}\nearrow. Hence in particular there is a lift when pKp \in K and so iKProji \in K Proj. The other case is formally dual.

closure under composition

Given a commuting square of the form

A X KInj p 1 K i KInj p 2 B Y \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

A X KInj p 1 K i KInj p 2 B Y. \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

A X K i KInj p 1 KInj p 2 B Y \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 1p 1p_1\circ p_1 has the right lifting property against KK and is hence in KInjK Inj. The case of composing two morphisms in KProjK Proj is formally dual. From this the closure of KProjK 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 jj be the retract of an iKProji \in K Proj, i.e. let there be a commuting diagram of the form.

id A: A C A j KProj i j id B: B D B. \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

A X j K f B Y \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

A C A X j KProj i j K f B D B Y. \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:

A C A X j i K f B D B Y. \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 jj has the left lifting property against all pKp \in K and hence is in KProjK Proj. The other case is formally dual.

closure under pushout and pullback

Let pKInjp \in K Inj and and let

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

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

A Z× fX K i f *p B g Z \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

A Z× fX X i f *p p B g Z f Y. \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 pp, there is a diagonal lift of the total outer diagram

A X i (fg)^ p B fg Y. \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 g^\hat g in

Z× fX X g^ f *p p B g Z f Y. \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 g^\hat g to qualify as the intended lift of the total diagram, it remains to show that

A Z× fX i g^ B \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 AA:

  • one is given by the morphisms

    1. AZ× fXXA \to Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z

    with universal morphism into the pullback being

    • AZ× fXA \to Z \times_f X
  • the other by

    1. AiBg^Z× fXXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z.

    with universal morphism into the pullback being

    • AiBg^Z× fXA \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 si sB s)KProj} sS\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S} be a set of elements of KProjK Proj. Since colimits in the presheaf category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects sSA s\underset{s \in S}{\coprod} A_s induced via its universal property by the set of morphisms i si_s:

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

Now let

sSA s X (i s) sS K f sSB s Y \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

{A s X KProj i s K f B s Y} sS. \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 s\ell_s. The collection of these lifts

{A s X Proj i s s K f B s Y} sS \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 ( s) sS(\ell_s)_{s\in S} in the original square

sSA s X (i s) sS ( s) sS K f sSB s Y. \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 si_s has the left lifting property against all fKf\in K and is hence in KProjK Proj. The other case is formally dual.

An immediate consequence of prop. 6 is this:

Corollary

Let 𝒞\mathcal{C} be a category with all small colimits, and let KMor(𝒞)K\subset Mor(\mathcal{C}) be a sub-class of its morphisms. Then every KK-injective morphism, def. 21, has the right lifting property, def. 21, against all KK-relative cell complexes, def. 15 and their retracts, remark 11.

Remark

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

id g:gfg. id_g \;\colon\; g \longrightarrow f \longrightarrow g \,.

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

id A: A X A g f g id B: B Y B. \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 CC the class of isomorphisms is preserved under retracts in the sense of remark 11.

Proof

For

id A: A X A g f g id B: B Y B. \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 XfYX \overset{f}{\to} Y an isomorphism, the inverse to AgBA \overset{g}{\to} B is given by the composite

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

More generally:

Proposition

Given a model category in the sense of def. 20, then its class of weak equivalences is closed under forming retracts (in the arrow category, see remark 11).

(Joyal, prop. E.1.3)

Proof

Let

id: A X A f w f id: B Y B \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 wWw \in W a weak equivalence. We need to show that then also fWf \in W.

First consider the case that fFibf \in Fib.

In this case, factor ww as a cofibration followed by an acyclic fibration. Since wWw \in W and by two-out-of-three (def. 19) this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain a commuting diagram of the following form:

id: A X AAAA A id WCof id id: A s X AAtAA A Fib f WFib Fib f id: B Y AAAA B, \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 ss is uniquely defined and where tt is any lift of the top middle vertical acyclic cofibration against ff. This now exhibits ff as a retract of an acyclic fibration. These are closed under retract by prop. 6.

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

id: A X AAAA A WCof (po) WCof WCof id: A X AAAA A Fib W Fib id: B Y AAAA B, \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 AA'. Since acyclic cofibrations are closed under forming pushouts by prop. 6, 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:XiApY. f \;\colon\; X \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y \,.
  1. If ff has the left lifting property against pp, then ff is a retract of ii.

  2. If ff has the right lifting property against ii, then ff is a retract of pp.

Proof

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

Write the factorization of ff as a commuting square of the form

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

By the assumed lifting property of ff against pp there exists a diagonal filler gg making a commuting diagram of the form

X i A f g p Y = Y. \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

X = X f i id Y: Y g A p Y. \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 11:

id X: X = X = X f i f id Y: Y g A p Y. \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 CMor(𝒞)C \subset Mor(\mathcal{C}) of morphisms in some category 𝒞\mathcal{C}, a natural question is how to factor any given morphism f:XYf\colon X \longrightarrow Y through a relative CC-cell complex, def. 15, followed by a CC-injective morphism, def. 17

f:XCcellX^CinjY. 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 X^=X 1\hat X = X_1 by attaching all possible CC-cells to XX. Namely let

(C/f){dom(c) X cC f cod(c) Y} (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 CC-cell attachment in ff, and consider the pushout X^\hat X of the coproduct of morphisms in CC over all these:

c(C/f)dom(c) X c(C/f)c (po) c(C/f)cod(c) X 1. \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 XX 1X \to X_1 is a CC-relative cell complex, by construction.

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

c(C/f)dom(c) X c(C/f)c f c(C/f)cod(c) Y \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 ff is indeed factored through that CC-cell complex X 1X_1; we may suggestively arrange that factorizing diagram like so:

c(C/f)dom(c) X id c(C/f)dom(c) X 1 c(C/f)c c(C/f)cod(c) Y. \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 1YX_1 \to Y against the cCc\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 cCc\in C into ff, but only those where the top morphism dom(c)X 1dom(c) \to X_1 factors through XX 1X \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 1YX_1 \to Y in the same way into

X 1X 2Y X_1 \longrightarrow X_2 \longrightarrow Y

and so forth. Since relative CC-cell complexes are closed under composition, at stage nn the resulting XX nX \longrightarrow X_n is still a CC-cell complex, getting bigger and bigger. But accordingly, the failure of the accompanying X nYX_n \longrightarrow Y to be a CC-injective morphism becomes smaller and smaller, for it now lifts against all diagrams where dom(c)X ndom(c) \longrightarrow X_n factors through X n1X nX_{n-1}\longrightarrow X_n, which intuitively is less and less of a condition as the X n1X_{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 𝒞\mathcal{C} a category and CMor(𝒞)C \subset Mor(\mathcal{C}) a sub-set of its morphisms, say that these have small domains if there is an ordinal α\alpha (def. 3) such that for every cCc\in C and for every CC-relative cell complex given by a transfinite composition (def. 4)

f:XX 1X 2X βX^ f \;\colon\; X \to X_1 \to X_2 \to \cdots \to X_\beta \to \cdots \longrightarrow \hat X

every morphism dom(c)X^dom(c)\longrightarrow \hat X factors through a stage X βX^X_\beta \to \hat X of order β<α\beta \lt \alpha:

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

The above discussion proves the following:

Proposition

(small object argument)

Let 𝒞\mathcal{C} be a locally small category with all small colimits. If a set CMor(𝒞)C\subset Mor(\mathcal{C}) of morphisms has all small domains in the sense of def. 25, then every morphism f:Xf\colon X\longrightarrow in 𝒞\mathcal{C} factors through a CC-relative cell complex, def. 15, followed by a CC-injective morphism, def. 17

f:XCcellX^CinjY. 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. 20.

Definition

Let 𝒞\mathcal{C} be a model category, def. 20, and X𝒞X \in \mathcal{C} an object.

  • A path space object Path(X)Path(X) for XX is a factorization of the diagonal Δ X:XX×X\Delta_X \;\colon\; X \to X \times X as
Δ X:XWiPath(X)Fib(p 0,p 1)X×X. \Delta_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \underoverset{\in Fib}{(p_0,p_1)}{\longrightarrow} X \times X \,.

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

  • A cylinder object Cyl(X)Cyl(X) for XX is a factorization of the codiagonal (or “fold map”) X:XXX\nabla_X \;\colon\; X \sqcup X \to X as
X:XXCof(i 0,i 1)Cyl(X)WpX. \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)XCyl(X) \to X is a weak equivalence. and XXCyl(X)X \sqcup X \to Cyl(X) is a cofibration.

Remark

For every object X𝒞X \in \mathcal{C} in a model category, a cylinder object and a path space object according to def. 26 exist: the factorization axioms guarantee that there exists

  1. a factorization of the codiagonal as

    X:XXCofCyl(X)WFibX \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

    Δ X:XWCofPath(X)FibX×X. \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. 26: in addition to Cyl(X)XCyl(X)\to X being just a weak equivalence, for these this is actually an acyclic fibration, and dually in addition to XPath(X)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. 26: what mainly matters is the weak equivalence, hence some authors take cylinder/path-space objects to be defined as in def. 26 but without the condition that XXCyl(X)X \sqcup X\to Cyl(X) is a cofibration and without the condition that Path(X)XPath(X) \to X is a fibration. Such authors would then refer to the concept in def. 26 as “good” cylinder/path-space objects.

The terminology in def. 26 follows the original (Quillen 67, I.1 def. 4). With the induced concept of left/right homotopy below in def. 27, this admits a quick derivation of the key facts in the following, as we spell out below.

Lemma

Let 𝒞\mathcal{C} be a model category. If X𝒞X \in \mathcal{C} is cofibrant, then for every cylinder object Cyl(X)Cyl(X) of XX, def. 26, not only is (i 0,i 1):XXX(i_0,i_1) \colon X \sqcup X \to X a cofibration, but each

i 0,i 1:XCyl(X) i_0, i_1 \colon X \longrightarrow Cyl(X)

is an acyclic cofibration separately.

Dually, if X𝒞X \in \mathcal{C} is fibrant, then for every path space object Path(X)Path(X) of XX, def. 26, not only is (p 0,p 1):Path(X)X×X(p_0,p_1) \colon Path(X)\to X \times X a cofibration, but each

p 0,p 1:Path(X)X 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)XPath(X) \to X and so this follows by two-out-of-three (def. 19).

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

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

hence p i:Path(X)X×XXp_i \colon Path(X)\to X \times X \to X is the composite of two fibrations, and hence itself a fibration, by prop. 6.

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

Example

If X𝒞X \in \mathcal{C} is a fibrant object in a model category, def. 20, and for Path 1(X)Path_1(X) and Path 2(X)Path_2(X) two path space objects for XX, def. 26, then the fiber product Path 1(X)× XPath 2(X)Path_1(X) \times_X Path_2(X) is another path space object for XX: the pullback square

X Δ X X×X Path 1(X)×XPath 2(X) Path 1(X)×Path 2(X) Fib (pb) Fib X×X×X (id,Δ X,id) X×X×X×X Fib (pr 1,pr 3) (p 1,p 4) X×X = X×X \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 8 and two-out-of-three (def. 19) gives that XPath 1(X)× XPath 2(X)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 11, with Path 1(X)=Path 2(X)=X I=X [0,1]Path_1(X) = Path_2(X) = X^I = X^{[0,1]} then this new path space object is X II=X [0,2]X^{I \vee I} = X^{[0,2]}, the mapping space out of the standard interval of length 2 instead of length 1.

Definition

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

  • A left homotopy η:f Lg\eta \colon f \Rightarrow_L g is a morphism η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y from a cylinder object of XX, def. 26, such that it makes this diagram commute:
X Cyl(X) X f η g Y. \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,.
  • A right homotopy η:f Rg\eta \colon f \Rightarrow_R g is a morphism η:XPath(Y)\eta \colon X \to Path(Y) to some path space object of XX, def. 26, such that this diagram commutes:
X f η g Y Path(Y) Y. \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,.
Lemma

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

  1. Let XX be cofibrant. If there is a left homotopy f Lgf \Rightarrow_L g then there is also a right homotopy f Rgf \Rightarrow_R g (def. 27) with respect to any chosen path space object.

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

In particular if XX is cofibrant and YY 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 η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y be the given left homotopy. Lemma 8 implies that we have a lift hh in the following commuting diagram

X if Path(Y) WCof i 0 h Fib p 0,p 1 Cyl(X) (fp,η) Y×Y, \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 η˜hi 1\tilde \eta \coloneqq h \circ i_1 is a right homotopy as required:

Path(Y) h Fib p 0,p 1 X i 1 Cyl(X) (fp,η) Y×Y. \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 XX a cofibrant object in a model category and YY a fibrant object, then the relations of left homotopy f Lgf \Rightarrow_L g and of right homotopy f Rgf \Rightarrow_R g (def. 27) on the hom set Hom(X,Y)Hom(X,Y) coincide and are both equivalence relations.

Proof

That both relations coincide under the (co-)fibrancy assumption follows directly from lemma 9.

The symmetry and reflexivity of the relation is obvious.

That right homotopy (hence also left homotopy) with domain XX is a transitive relation follows from using example 18 to compose path space objects.

The homotopy category

We discuss the construction that takes a model category, def. 20, and then universally forces all its weak equivalences into actual isomorphisms.

Definition

Let 𝒞\mathcal{C} be a model category, def. 20. Write Ho(𝒞)Ho(\mathcal{C}) for the category whose

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

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

Proposition

Def. 28 is well defined, in that composition of morphisms between fibrant-cofibrant objects in 𝒞\mathcal{C} indeed passes to homotopy classes.

Proof

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

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

to be well defined, we need that with (gh):YZ(g\sim h)\;\colon\; Y \to Z also (fgfh):XZ(f g \sim f h)\;\colon\; X \to Z. But by prop 10 we may take the homotopy \sim to be exhibited by a right homotopy η:YPath(Z)\eta \colon Y \to Path(Z), for which case the statement is evident from this diagram:

Z g p 1 X f Y η Path(Z) h p 0 Z. \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. 28 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 𝒞\mathcal{C} be a model category. A weak equivalence between two objects which are both fibrant and cofibrant is a homotopy equivalence.

Proof

By the factorization axioms in the model category 𝒞\mathcal{C} and by two-out-of-three (def. 19), every weak equivalence f:XYf\colon X \longrightarrow Y factors through an object ZZ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with XX and YY both fibrant and cofibrant, so is ZZ, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.

So let f:XYf \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 1f^{-1} in the diagram

X cof f 1 FibW f X = X. \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 1f^{-1} is also a left inverse up to left homotopy, let Cyl(X)Cyl(X) be any cylinder object on XX (def. 26), hence a factorization of the codiagonal on XX as a cofibration followed by a an acyclic fibration

XXι XCyl(X)pX X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{p}{\longrightarrow} X

and consider the commuting square

XX (f 1f,id) X ι X Cof WFib f Cyl(X) fp Y, \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 1f^{-1} being a genuine right inverse of ff. By construction, this commuting square now admits a lift η\eta, and that constitutes a left homotopy η:f 1f Lid\eta \colon f^{-1}\circ f \Rightarrow_L id.

Definition

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

  1. a factorization Cofi XQXWFibp XX\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_X}{\longrightarrow} X of the initial morphism, such that when XX is already cofibrant then p X=id Xp_X = id_X;

  2. a factorization XWCofj XPXFibq X*X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} P X \underoverset{\in Fib}{q_X}{\longrightarrow} \ast of the terminal morphism, such that when XX is already fibrant then j X=id Xj_X = id_X.

Write then

γ P,Q:𝒞Ho(𝒞) \gamma_{P,Q} \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})

for the functor to the homotopy category, def. 28, which sends an object XX to the object PQXP Q X and sends a morphism f:XYf \colon X \longrightarrow Y to the homotopy class of the result of first lifting in

QY i X Qf p Y QX fp X Y \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

QX j QYQf PQY j QX PQf q QY PQX *. \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. 29 is indeed well defined.

Proof

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

Cof Cof QX WCof PQX Fib * W 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 (Qf) i(Q f)_{i} of the first lift in the definition are left homotopic to each other, exhibited by lifting in

QXQX ((Qf) 1,(Qf) 2) QY Cof WFib p Y Cyl(QX) fp Xσ QX Y. \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 QY(Q f) ij_{Q Y}\circ (Q_f)_i are left homotopic to each other, and since their domain is cofibrant, then by lemma 9 they are also right homotopic by a right homotopy κ\kappa. This implies finally, by lifting in

QX κ Path(PQY) WCof Fib PQX (R(Qf) 1,P(Qf) 2) PQY×PQY \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(Qf) 1P (Q f)_1 and P(Qf) 2P (Q f)_2 are right homotopic, hence that indeed PQfP 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 QfQ f and PQfP Q f imply that also the following diagram commutes

X p X QX j QX PQX f Qf PQf Y p y QY j QY PQY. \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

X p X QX j QX PQX f Qf PQf Y p Y QY j QY PQY g Qg PQg Z p Z QZ j QZ PQZ \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 (PQg)(PQf)(P Q g)\circ (P Q f) is a lift of gfg \circ f and hence the same argument as above gives that it is homotopic to the chosen PQ(gf)P Q(g \circ f).

For the following, recall the concept of natural isomorphism between functors: for F,G:𝒞𝒟F, G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} two functors, then a natural transformation η:FG\eta \colon F \Rightarrow G is for each object cObj(𝒞)c \in Obj(\mathcal{C}) a morphism η c:F(c)G(c)\eta_c \colon F(c) \longrightarrow G(c) in 𝒟\mathcal{D}, such that for each morphism f:c 1c 2f \colon c_1 \to c_2 in 𝒞\mathcal{C} the following is a commuting square:

F(c 1) η c 1 G(c 1) F(f) G(f) F(c 2) η c 2 G(c 2). \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 η\eta is called a natural isomorphism if its η c\eta_c are isomorphisms for all objects cc.

Definition

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

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

  2. a functor

    γ:𝒞𝒞[W 1] \gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]

such that

  1. γ\gamma sends weak equivalences to isomorphisms;

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

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

    𝒞 F D γ ρ F˜ Ho(𝒞) \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 (F˜ 1,ρ 1)(\tilde F_1, \rho_1) and (F˜ 2,ρ 2)(\tilde F_2, \rho_2) two such factorizations, then there is a unique natural isomorphism κ:F˜ 1F˜ 2\kappa \colon \tilde F_1 \Rightarrow \tilde F_2 making the evident diagram of natural isomorphisms commute.

Theorem

For 𝒞\mathcal{C} a model category, the functor γ P,Q\gamma_{P,Q} in def. 29 (for any choice of PP and QQ) exhibits Ho(𝒞)Ho(\mathcal{C}) as indeed being the localization of the underlying category with weak equivalences at its weak equivalences, in the sense of def. 30:

𝒞 = 𝒞 γ P,Q γ Ho(𝒞) 𝒞[W 1]. \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 γ P,Q\gamma_{P,Q} indeed takes weak equivalences to isomorphisms: By two-out-of-three (def. 19) applied to the commuting diagrams shown in the proof of lemma 11, the morphism PQfP Q f is a weak equivalence if ff is:

X p X QX j QX PQX f Qf PQf Y p y QY j QY PQY \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 10, implies that PQfP Q f represents an isomorphism in Ho(𝒞)Ho(\mathcal{C}).

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

𝒞 F D γ ρ F˜ Ho(𝒞) \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 PP and QQ in def. 29, γ P,Q\gamma_{P,Q} is the identity on the full subcategory of fibrant-cofibrant objects. It follows that if F˜\tilde F exists at all, it must satisfy for all XfYX \stackrel{f}{\to} Y with XX and YY both fibrant and cofibrant that

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

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

But by def. 28 that already fixes F˜\tilde F on all of Ho(𝒞)Ho(\mathcal{C}), up to unique natural isomorphism. Hence it only remains to check that with this definition of F˜\tilde F there exists any natural isomorphism ρ\rho filling the diagram above.

To that end, apply FF to the above commuting diagram to obtain

F(X) isoF(p X) F(QX) isoF(j QX) F(PQX) F(f) F(Qf) F(PQf) F(Y) F(p y)iso F(QY) F(j QY)iso F(PQY). \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 FF. It follows that defining ρ XF(j QX)F(p X) 1\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1} makes the required natural isomorphism:

ρ X: F(X) isoF(p X) 1 F(QX) isoF(j QX) F(PQX) = F˜(γ P,Q(X)) F(f) F(PQf) F˜(γ P,Q(f)) ρ Y: F(Y) F(p y) 1iso F(QY) F(j QY)iso F(PQY) = F˜(γ P,Q(X)). \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 1 we may suppress the choices of cofibrant QQ and fibrant replacement PP in def. 29 and just speak of the localization functor

γ:𝒞Ho(𝒞) \gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})

up to natural isomorphism.

In general, the localization 𝒞[W 1]\mathcal{C}[W^{-1}] of a category with weak equivalences (𝒞,W)(\mathcal{C},W) (def. 30) may invert more morphisms than just those in WW. However, if the category admits the structure of a model category (𝒞,W,Cof,Fib)(\mathcal{C},W,Cof,Fib), then its localiztion precisely only inverts the weak equivalences.

Proposition

Let 𝒞\mathcal{C} be a model category (def. 20) and let γ:𝒞Ho(𝒞)\gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) be its localization functor (def. 29, theorem 1). Then a morphism ff in 𝒞\mathcal{C} is a weak equivalence precisely if γ(f)\gamma(f) is an isomorphism in Ho(𝒞)Ho(\mathcal{C}).

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

While the construction of the homotopy category in def. 28 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 𝒞\mathcal{C}, write

𝒞 fc 𝒞 c 𝒞 f 𝒞 \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 𝒞 f\mathcal{C}_f

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

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

all regarded a categories with weak equivalences (def. 19), via the weak equivalences inherited from 𝒞\mathcal{C}, which we write (𝒞 f,W f)(\mathcal{C}_f, W_f), (𝒞 c,W c)(\mathcal{C}_c, W_c) and (𝒞 fc,W fc)(\mathcal{C}_{f c}, W_{f c}).

Remark

Of course the subcategories in def. 31 inherit more structure than just that of categories with weak equivalences from 𝒞\mathcal{C}. 𝒞 f\mathcal{C}_f and 𝒞 c\mathcal{C}_c each inherit “half” of the factorization axioms. One says that 𝒞 f\mathcal{C}_f has the structure of a “fibration category” called a “Brown-category of fibrant objects”, while 𝒞 c\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 1 immediately implies the following:

Corollary

For 𝒞\mathcal{C} a model category, the restriction of the localization functor γ:𝒞Ho(𝒞)\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) from def. 29 (using remark 13) to any of the sub-categories with weak equivalences of def. 31

𝒞 fc 𝒞 c 𝒞 f 𝒞 γ Ho(𝒞) \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(𝒞)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(𝒞)𝒞[W 1]𝒞 f[W f 1]𝒞 c[W c 1]𝒞 fc[W fc 1]. 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

For X,Y𝒞X, Y \in \mathcal{C} with XX cofibrant and YY fibrant, and for P,QP, Q fibrant/cofibrant replacement functors as in def. 29, then the morphism

Hom Ho(𝒞)(PX,QY)=Hom 𝒞(PX,QY)/ Hom 𝒞(j X,p Y)Hom 𝒞(X,Y)/ 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. 10) is a natural bijection.

(Quillen 67, I.1 lemma 7)

Proof

We may factor the morphism in question as the composite

Hom 𝒞(PX,QY)/ Hom 𝒞(id PX,p Y)/ Hom 𝒞(PX,Y)/ Hom 𝒞(j X,id Y)/ Hom 𝒞(X,Y)/ . 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 XX cofibrant and YY fibrant, then

Hom 𝒞(id X,p Y)/ :Hom 𝒞(X,QY)/ Hom 𝒞(X,Y)/ 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 𝒞(j X,id Y)/ :Hom 𝒞(PX,Y)/ Hom 𝒞(X,Y)/ 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 𝒞(id X,p Y)Hom_{\mathcal{C}}(id_X, p_Y) is surjective is the lifting property in

QY Cof WFib p Y X f Y, \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:XYf \colon X \to Y comes from a morphism f^:XQY\hat f \colon X \to Q Y under postcomposition with QYp YYQ Y \overset{p_Y}{\to} Y.

Second, that Hom 𝒞(id X,p Y)Hom_{\mathcal{C}}(id_X, p_Y) is injective is the lifting property in

XX (f,g) QY Cof WFib p Y Cyl(X) η Y, \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:XQYf, g \colon X \to Q Y become homotopic after postcomposition with p Y:QXYp_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 𝒞\mathcal{C} be a model category (def. 20). Then every commuting square in its homotopy category Ho(C)Ho(C) (def. 28) is, up to isomorphism of squares, in the image of the localization functor 𝒞Ho(𝒞)\mathcal{C} \longrightarrow Ho(\mathcal{C}) of a commuting square in 𝒞\mathcal{C} (i.e.: not just commuting up to homotopy).

Proof

Let

A f B a b A f BHo(𝒞) \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 𝒞\mathcal{C} and for morphisms in 𝒞\mathcal{C} representing these, then this means that in 𝒞\mathcal{C} there is a left homotopy of the form

A f B i 1 b Cyl(A) η B i 0 f A a A. \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 ff

A f B i 1 (po) W Cyl(A) Cyl(f) i 0 η A B a f A \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 Ai 0Cyl(A)Cyl(f)A \overset{i_0}{\to} Cyl(A) \to Cyl(f) as an alternative representative of ff in Ho(𝒞)Ho(\mathcal{C}), and Cyl(f)BCyl(f) \to B' as an alternative representative for bb, and the commuting square

A Cyl(f) a A f B \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(𝒞)Ho(\mathcal{C}).

Derived functors

Definition

For 𝒞\mathcal{C} and 𝒟\mathcal{D} two categories with weak equivalences, def. 19, then a functor F:𝒞𝒟F \colon \mathcal{C}\longrightarrow \mathcal{D} is called a homotopical functor if it sends weak equivalences to weak equivalences.

Definition

Given a homotopical functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} (def. 32) between categories with weak equivalences whose homotopy categories Ho(𝒞)Ho(\mathcal{C}) and Ho(𝒟)Ho(\mathcal{D}) exist (def. 30), then its (“total”) derived functor is the functor Ho(F)Ho(F) between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. 30):

𝒞 F 𝒟 γ 𝒞 γ 𝒟 Ho(𝒞) Ho(F) Ho(𝒟). \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. 32, many become homotopical after restriction to the full subcategories 𝒞 f\mathcal{C}_f of fibrant objects or 𝒞 c\mathcal{C}_c of cofibrant objects, def. 31. By corollary 1 this is just as good for the purpose of homotopy theory.

Therefore one considers the following generalization of def. 33:

Definition

Consider a functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} out of a model category 𝒞\mathcal{C} (def. 20) into a category with weak equivalences 𝒟\mathcal{D} (def. 19).

  1. If the restriction of FF to the full subcategory 𝒞 f\mathcal{C}_f of fibrant object becomes a homotopical functor (def. 32), then the derived functor of that restriction, according to def. 33, is called the right derived functor of FF and denoted by F\mathbb{R}F:

    𝒞 f 𝒞 F 𝒟 γ 𝒞 f γ 𝒟 F: 𝒞 f[W 1] Ho(𝒞) Ho(F) Ho(𝒟), \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 1.

  2. If the restriction of FF to the full subcategory 𝒞 c\mathcal{C}_c of cofibrant object becomes a homotopical functor (def. 32), then the derived functor of that restriction, according to def. 33, is called the left derived functor of FF and denoted by 𝕃F\mathbb{L}F:

    𝒞 c 𝒞 F 𝒟 γ 𝒞 f γ 𝒟 𝕃F: 𝒞 c[W 1] Ho(𝒞) Ho(F) Ho(𝒟), \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 1.

The key fact that makes def. 34 practically relevant is the following:

Proposition

(Ken Brown's lemma)

Let 𝒞\mathcal{C} be a model category with full subcategories 𝒞 f,𝒞 c\mathcal{C}_f, \mathcal{C}_c of fibrant objects and of cofibrant objects respectively (def. 31). Let 𝒟\mathcal{D} be a category with weak equivalences.

  1. A functor out of the category of fibrant objects

    F:𝒞 f𝒟 F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D}

    is a homotopical functor, def. 32, already if it sends acylic fibrations to weak equivalences.

  2. A functor out of the category of cofibrant objects

    F:𝒞 c𝒟 F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D}

    is a homotopical functor, def. 32, already if it sends acylic cofibrations to weak equivalences.

The following proof refers to the factorization lemma, whose full statement and proof we postpone to further below (lemma 19).

Proof

We discuss the case of a functor on a category of fibrant objects 𝒞 f\mathcal{C}_f, def. 31. The other case is formally dual.

Let f:XYf \colon X \longrightarrow Y be a weak equivalence in 𝒞 f\mathcal{C}_f. Choose a path space object Path(X)Path(X) (def. 26) and consider the diagram

Path(f) WFib X W p 1 *f (pb) W f Path(Y) WFibp 1 Y WFib p 0 Y, \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)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 ff, def. 58).

Here:

  1. p ip_i are both acyclic fibrations, by lemma 8;

  2. Path(f)XPath(f) \to X is an acyclic fibration because it is the pullback of p 1p_1.

  3. p 1 *fp_1^\ast f is a weak equivalence, because the factorization lemma 19 states that the composite vertical morphism factors ff through a weak equivalence, hence if ff is a weak equivalence, then p 1 *fp_1^\ast f is by two-out-of-three (def. 19).

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

F(Path(f)) W F(X) F(p 1 *f) F(f) F(Path(Y)) WF(p 1) F(Y) W F(p 0) Y. \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 19, in addition says that the vertical composite p 0p 1 *fp_0 \circ p_1^\ast f is a fibration, hence an acyclic fibration by the above. Therefore also F(p 0p 1 *f)F(p_0 \circ p_1^\ast f) is a weak equivalence. Now the claim that also F(f)F(f) is a weak equivalence follows with applying two-out-of-three (def. 19) twice.

Corollary

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

  1. If FF preserves cofibrant objects and acyclic cofibrations between these, then its left derived functor (def. 34) 𝕃F\mathbb{L}F exists, fitting into a diagram

    𝒞 c F 𝒟 c γ 𝒞 γ 𝒟 Ho(𝒞) 𝕃F Ho(𝒟) \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 FF preserves fibrant objects and acyclic fibrants between these, then its right derived functor (def. 34) F\mathbb{R}F exists, fitting into a diagram

    𝒞 f F 𝒟 f γ 𝒞 γ 𝒟 Ho(𝒞) F Ho(𝒟). \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

Let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a functor between two model categories (def. 20).

  1. If FF preserves fibrant objects and weak equivalences between fibrant objects, then the total right derived functor F(γ 𝒟F)\mathbb{R}F \coloneqq \mathbb{R}(\gamma_{\mathcal{D}}\circ F) (def. 34) in

    𝒞 f F 𝒟 γ 𝒞 f γ 𝒟 Ho(𝒞) F Ho(𝒟) \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 X𝒞γ 𝒞Ho(𝒞) X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C}) by appying FF to a fibrant replacement PXP X of XX and then forming a cofibrant replacement Q(F(PX))Q(F(P X)) of the result:

F(X)Q(F(PX)). \mathbb{R}F(X) \simeq Q(F(P X)) \,.
  1. If FF preserves cofibrant objects and weak equivalences between cofibrant objects, then the total left derived functor 𝕃F𝕃(γ 𝒟F)\mathbb{L}F \coloneqq \mathbb{L}(\gamma_{\mathcal{D}}\circ F) (def. 34) in

    𝒞 c F 𝒟 γ 𝒞 c γ 𝒟 Ho(𝒞) 𝕃F Ho(𝒟) \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 X𝒞γ 𝒞Ho(𝒞) X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C}) by appying FF to a cofibrant replacement QXQ X of XX and then forming a fibrant replacement P(F(QX))P(F(Q X)) of the result:

𝕃F(X)P(F(QX)). \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 1 we have

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

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

Now since FF is assumed to preserve fibrant objects, F(P(X))F(P(X)) is fibrant in 𝒟\mathcal{D}, and hence γ 𝒟\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 2 to be satisfied by pairs of adjoint functors. Recall that this is a pair of functors LL and RR going back and forth between two categories

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

such that there is a natural bijection between hom-sets with LL on the left and those with RR on the right:

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

for all objects d𝒟d\in \mathcal{D} and c𝒞c \in \mathcal{C}. This being natural means that ϕ:Hom 𝒟(L(),)Hom 𝒞(,R())\phi \colon Hom_{\mathcal{D}}(L(-),-) \Rightarrow Hom_{\mathcal{C}}(-, R(-)) is a natural transformation, hence that for all morphisms g:d 2d 1g \colon d_2 \to d_1 and f:c 1c 2f \colon c_1 \to c_2 the following is a commuting square:

Hom 𝒞(L(d 1),c 1) ϕ d 1,c 1 Hom 𝒟(d 1,R(c 1)) L(f)()g g()R(g) Hom 𝒞(L(d 2),c 2) ϕ d 2,c 2 Hom 𝒟(d 2,R(c 2)). \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 (LR)(L \dashv R) to indicate an adjunction and call LL the left adjoint and RR the right adjoint of the adjoint pair.

The archetypical example of a pair of adjoint functors is that consisting of forming Cartesian products Y×()Y \times (-) and forming mapping spaces () Y(-)^Y, as in the category of compactly generated topological spaces of def. 46.

If f:L(d)cf \colon L(d) \to c is any morphism, then the image ϕ d,c(f):dR(c)\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

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

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

For an object c𝒞c \in \mathcal{C}, the adjunct of the identity on RcR c is called the adjunction counit ϵ c:LRcc\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

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

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

Definition

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

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

is called a Quillen adjunction (and LL,RR are called left/right Quillen functors, respectively) if the following equivalent conditions are satisfied

  1. LL preserves cofibrations and RR preserves fibrations;

  2. LL preserves acyclic cofibrations and RR preserves acyclic fibrations;

  3. LL preserves cofibrations and acylic cofibrations;

  4. RR preserves fibrations and acyclic fibrations.

Proposition

The conditions in def. 35 are indeed all equivalent.

(Quillen 67, I.4, theorem 3)

Proof

First observe that

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

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

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

Lemma

Let 𝒞RL𝒟\mathcal{C} \stackrel{\overset{L}{\longleftarrow}}{\underoverset{R}{\bot}{\longrightarrow}} \mathcal{D} be a Quillen adjunction, def. 35.

  1. For X𝒞X \in \mathcal{C} a fibrant object and Path(X)Path(X) a path space object (def. 26), then R(Path(X))R(Path(X)) is a path space object for R(X)R(X).

  2. For X𝒞X \in \mathcal{C} a cofibrant object and Cyl(X)Cyl(X) a cylinder object (def. 26), then L(Cyl(X))L(Cyl(X)) is a path space object for L(X)L(X).

Proof

Consider the second case, the first is formally dual.

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

Hence

L(XXCofCyl(X))=(L(X)L(X)CofL(Cyl(X))) 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 YY cofibrant then also YCyl(Y)Y \sqcup Cyl(Y) is a cofibrantion, since YYYY \to Y \sqcup Y is a cofibration (lemma 8). Therefore by Ken Brown's lemma (prop. 13) LL preserves the weak equivalence Cyl(Y)WYCyl(Y) \overset{\in W}{\longrightarrow} Y.

Proposition

For 𝒞RL𝒟\mathcal{C} \underoverset{\underoverset{R}{\bot}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{D} a Quillen adjunction, def. 35, then also the corresponding left and right derived functors, def. 34, via cor. 2, form a pair of adjoint functors

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

(Quillen 67, I.4 theorem 3)

Proof

By def. 34 and lemma 12 it is sufficient to see that for X,Y𝒞X, Y \in \mathcal{C} with XX cofibrant and YY fibrant, then there is a natural bijection

Hom 𝒞(LX,Y)/ Hom 𝒞(X,RY)/ . Hom_{\mathcal{C}}(L X , Y)/_\sim \simeq Hom_{\mathcal{C}}(X, R Y)/_\sim \,.

Since by the adjunction isomorphism for (LR)(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 14 that with Cyl(Y)Cyl(Y) a cylinder object for YY, def. 26, then L(Cyl(Y))L(Cyl(Y)) is a cylinder object for L(Y)L(Y). This implies that left homotopies

(f Lg):LXY (f \Rightarrow_L g) \;\colon\; L X \longrightarrow Y

given by

η:Cyl(LX)=LCyl(X)Y \eta \;\colon\; Cyl(L X) = L Cyl(X) \longrightarrow Y

are in bijection to left homotopies

(f˜ Lg˜):XRY (\tilde f \Rightarrow_L \tilde g) \;\colon\; X \longrightarrow R Y

given by

η˜:Cyl(X)RX. \tilde \eta \;\colon\; Cyl(X) \longrightarrow R X \,.
Definition

For 𝒞,𝒟\mathcal{C}, \mathcal{D} two model categories, a Quillen adjunction (def.35)

(LR):𝒞RL𝒟 (L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

is called a Quillen equivalence, to be denoted

𝒞 QRL𝒟, \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\simeq_{\mathrlap{Q}}} \mathcal{D} \,,

if the following equivalent conditions hold.

  1. The right derived functor of RR (via prop. 14, corollary 2) is an equivalence of categories

    R:Ho(𝒞)Ho(𝒟). \mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D}) \,.
  2. The left derived functor of LL (via prop. 14, corollary 2) is an equivalence of categories

    𝕃L:Ho(𝒟)Ho(𝒞). \mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C}) \,.
  3. For every cofibrant object d𝒟d\in \mathcal{D}, the “derived adjunction unit”, hence the composite

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

    (of the adjunction unit with any fibrant replacement PP as in def. 29) is a weak equivalence;

    and for every fibrant object c𝒞c \in \mathcal{C}, the “derived adjunction counit”, hence the composite

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

    (of the adjunction counit with any cofibrant replacement as in def. 29) is a weak equivalence in DD.

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

    dW 𝒟R(c)L(d)W 𝒞c. \frac{ d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c) }{ L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c } \,.
Poposition

The conditions in def. 36 are indeed all equivalent.

(Quillen 67, I.4, theorem 3)

Proof

That 1)2)1) \Leftrightarrow 2) follows from prop. 15 (if in an adjoint pair one is an equivalence, then so is the other).

To see the equivalence 1),2)3)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 show that the morphisms called “derived adjunction (co-)units” above indeed represent the adjunction (co-)unit of (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) in the homotopy category. We show this now for the adjunction unit, the case of the adjunction counit is formally dual.

To that end, first observe that for d𝒟 cd \in \mathcal{D}_c, then the defining commuting square for the left derived functor from def. 34

𝒟 c L 𝒞 γ P γ P,Q Ho(𝒟) 𝕃L Ho(𝒞) \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 γ P\gamma_P, γ P,Q\gamma_{P,Q} from def. 29 with their universal property from theorem 1, corollary 1) gives that

(𝕃L)dPLPdPLdHo(𝒞), (\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 LL sends the acyclic cofibration j d:dPdj_d \colon d \to P d to a weak equivalence.

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

Hom Ho(𝒞)((𝕃L)Pd,(𝕃L)Pd)Hom Ho(𝒞)(Pd,(R)(𝕃L)Pd). 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. 15, that adjunction isomorphism is equivalently that of (LR)(L \dashv R) under the isomorphism

Hom Ho(𝒞)(PLd,PLd)Hom(j Ld,id)Hom 𝒞(Ld,PLd)/ 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 12. Hence the derived adjunction unit is the (LR)(L \dashv R)-adjunct of

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

which indeed (by the formula for adjuncts) is

XηRLdR(j Ld)RPLd. X \overset{\eta}{\longrightarrow} R L d \overset{R (j_{L d})}{\longrightarrow} R P L d \,.

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

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

XηRLXRj LXRPLX 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 unit in 3). Dually for derived counit.

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

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

f˜: d η RLd Rf Rc = Rj Ld Rj c d W RPLd RPf RPc, \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 PfP f is any lift constructed as in def. 29.

This exhibits the bottom left morphism as the derived adjunction unit, hence a weak equivalence by assumption. But since ff was a weak equivalence, so is PfP f (by two-out-of-three). Thereby also RPfR P f and Rj YR j_Y, are weak equivalences by Ken Brown's lemma 13 and the assumed fibrancy of cc. Therefore by two-out-of-three (def. 19) also the adjunct f˜\tilde f is a weak equivalence.

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

Proposition

If in a Quillen adjunction 𝒞 RL 𝒟 \array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}} (def. 35) the right adjoint RR “creates weak equivalences” (in that a morphism ff in 𝒞\mathcal{C} is a weak equivalence precisly if U(f)U(f) is) then (LR)(L \dashv R) is a Quillen equivalence (def. 36) precisely already if for all cofibrant objects d𝒟d \in \mathcal{D} the plain adjunction unit

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

is a weak equivalence.

Proof

By prop. 16, generally, (LR)(L \dashv R) is a Quillen equivalence precisely if

  1. for every cofibrant object d𝒟d\in \mathcal{D}, the “derived adjunction unit”

    dηR(L(d))R(j L(d))R(P(L(d))) 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 c𝒞c \in \mathcal{C}, the “derived adjunction counit”

    L(Q(R(c)))L(p R(c))L(R(c))ϵc 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 RR preserves the weak equivalence j L(d)j_{L(d)}, then by two-out-of-three (def. 19) the composite in the first item is a weak equivalence precisely if η\eta is.

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

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

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

under RR is.

Moreover, assuming, by the above, that η Q(R(c))\eta_{Q(R(c))} on the cofibrant object Q(R(c))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 η\eta is

Q(R(c))η Q(R(c))R(L(Q(R(c))))R(L(p R(c)))R(L(R(c)))R(ϵ)R(c). 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 (LR)(L\dashv R)-adjunct of the original composite, which is just p R(c)p_{R(c)}

L(Q(R(c)))L(p R(c))L(R(c))ϵcQ(R(C))p R(c)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)p_{R(c)} is a weak equivalence by definition of cofibrant replacement.

The model structure on topological spaces

We now discuss how the category Top of topological spaces satisfies the axioms of abstract homotopy theory (model category) theory, def. 20.

Definition

Say that a continuous function, hence a morphism in Top, is

and hence

  • a acyclic classical cofibration if it is a classical cofibration as well as a classical weak equivalence;

  • a acyclic classical fibration if it is a classical fibration as well as a classical weak equivalence.

Write

W cl,Fib cl,Cof clMor(Top) W_{cl},\;Fib_{cl},\;Cof_{cl} \subset Mor(Top)

for the classes of these morphisms, respectively.

We first prove now that the classes of morphisms in def. 37 satisfy the conditions for a model category structure, def. 20 (after some lemmas, this is theorem 2 below). Then we discuss the resulting classical homotopy category (below) and then a few variant model structures whose proof follows immediately along the line of the proof of Top QuillenTop_{Quillen}:

\,

Proposition

The classical weak equivalences, def. 37, satify two-out-of-three (def. 19).

Proof

Since isomorphisms (of homotopy groups) satisfy 2-out-of-3, this property is directly inherited via the very definition of weak homotopy equivalence, def. 10.

Lemma

Every morphism f:XYf\colon X \longrightarrow Y in Top factors as a classical cofibration followed by an acyclic classical fibration, def. 37:

f:XCof clX^W clFib clY. f \;\colon\; X \stackrel{\in Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in W_{cl} \cap Fib_{cl}}{\longrightarrow} Y \,.
Proof

By lemma 1 the set I Top={S n1D n}I_{Top} = \{S^{n-1}\hookrightarrow D^n\} of topological generating cofibrations, def. 13, has small domains, in the sense of def. 25 (the n-spheres are compact). Hence by the small object argument, prop. 9, ff factors as an I TopI_{Top}-relative cell complex, def. 15, hence just a plain relative cell complex, def. 14, followed by an I TopI_{Top}-injective morphisms, def. 17:

f:XCof clX^I TopInjY. f \;\colon\; X \stackrel{\in Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in I_{Top} Inj}{\longrightarrow} Y \,.

By lemma 5 the map X^Y\hat X \to Y is both a weak homotopy equivalence as well as a Serre fibration.

Lemma

Every morphism f:XYf\colon X \longrightarrow Y in Top factors as an acyclic classical cofibration followed by a fibration, def. 37:

f:XW clCof clX^Fib clY. f \;\colon\; X \stackrel{\in W_{cl} \cap Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in Fib_{cl}}{\longrightarrow} Y \,.
Proof

By lemma 1 the set J Top={D nD n×I}J_{Top} = \{D^n \hookrightarrow D^n\times I\} of topological generating acyclic cofibrations, def. 16, has small domains, in the sense of def. 25 (the n-disks are compact). Hence by the small object argument, prop. 9, ff factors as an J TopJ_{Top}-relative cell complex, def. 15, followed by a J topJ_{top}-injective morphisms, def. 17:

f:XJ TopCellX^J TopInjY. f \;\colon\; X \stackrel{\in J_{Top} Cell}{\longrightarrow} \hat X \stackrel{\in J_{Top} Inj}{\longrightarrow} Y \,.

By definition this makes X^Y\hat X \to Y a Serre fibration, hence a fibration.

By lemma 3 a relative J TopJ_{Top}-cell complex is in particular a relative I TopI_{Top}-cell complex. Hence XX^X \to \hat X is a classical cofibration. By lemma 4 it is also a weak homotopy equivalence, hence a clasical weak equivalence.

Lemma

Every commuting square in Top with the left morphism a classical cofibration and the right morphism a fibration, def. 37

gCof cl fFib cl \array{ &\longrightarrow& \\ {}^{\mathllap{{g \in} \atop {Cof_{cl}}}}\downarrow && \downarrow^{\mathrlap{{f \in }\atop Fib_{cl}}} \\ &\longrightarrow& }

admits a lift as soon as one of the two is also a classical weak equivalence.

Proof

A) If the fibration ff is also a weak equivalence, then lemma 5 says that it has the right lifting property against the generating cofibrations I TopI_{Top}, and cor. 7 implies the claim.

B) If the cofibration gg on the left is also a weak equivalence, consider any factorization into a relative J TopJ_{Top}-cell complex, def. 16, def. 15, followed by a fibration,

g:J TopCellFib cl, g \;\colon\; \stackrel{\in J_{Top} Cell}{\longrightarrow} \stackrel{\in Fib_{cl}}{\longrightarrow} \,,

as in the proof of lemma 16. By lemma 4 the morphism J TopCell\overset{\in J_{Top} Cell}{\longrightarrow} is a weak homotopy equivalence, and so by two-out-of-three (prop. 18) the factorizing fibration is actually an acyclic fibration. By case A), this acyclic fibration has the right lifting property against the cofibration gg itself, and so the retract argument, lemma 7 gives that gg is a retract of a relative J TopJ_{Top}-cell complex. With this, finally cor. 7 implies that ff has the right lifting property against gg.

Finally:

Proposition

The systems (Cof cl,W clFib cl)(Cof_{cl} , W_{cl} \cap Fib_{cl}) and (W clCof cl,Fib cl)(W_{cl} \cap Cof_{cl}, Fib_{cl}) from def. 37 are weak factorization systems.

Proof

Since we have already seen the factorization property (lemma 15, lemma 16) and the lifting properties (lemma 17), it only remains to see that the given left/right classes exhaust the class of morphisms with the given lifting property.

For the classical fibrations this is by definition, for the the classical acyclic fibrations this is by lemma 5.

The remaining statement for Cof clCof_{cl} and W clCof clW_{cl}\cap Cof_{cl} follows from a general argument (here) for cofibrantly generated model categories (def. 38), which we spell out:

So let f:XYf \colon X \longrightarrow Y be in (I TopInj)Proj(I_{Top} Inj) Proj, we need to show that then ff is a retract (remark 11) of a relative cell complex. To that end, apply the small object argument as in lemma 15 to factor ff as

f:XI TopCellY^I TopInjY. f \;\colon \; X \overset{I_{Top} Cell}{\longrightarrow} \hat Y \overset{\in I_{Top} Inj}{\longrightarrow} Y \,.

It follows that ff has the left lifting property against Y^Y\hat Y \to Y, and hence by the retract argument (lemma 7) it is a retract of XICellY^X \overset{I Cell}{\to} \hat Y. This proves the claim for Cof clCof_{cl}.

The analogous argument for W clCof clW_{cl} \cap Cof_{cl}, using the small object argument for J TopJ_{Top}, shows that every f(J TopInj)Projf \in (J_{Top} Inj) Proj is a retract of a J TopJ_{Top}-cell complex. By lemma 3 and lemma 4 a J TopJ_{Top}-cell complex is both an I TopI_{Top}-cell complex and a weak homotopy equivalence. Retracts of the former are cofibrations by definition, and retracts of the latter are still weak homotopy equivalences by lemma 6. Hence such ff is an acyclic cofibration.

In conclusion, prop. 18 and prop. 19 say that:

Theorem

The classes of morphisms in Mor(Top)Mor(Top) of def. 37,

define a model category structure (def. 20) Top QuillenTop_{Quillen}, the classical model structure on topological spaces or Serre-Quillen model structure .

In particular

  1. every object in Top QuillenTop_{Quillen} is fibrant;

  2. the cofibrant objects in Top QuillenTop_{Quillen} are the retracts of cell complexes.

Hence in particular the following classical statement is an immediate corollary:

Corollary

(Whitehead theorem)

Every weak homotopy equivalence (def. 10) between topological spaces that are homeomorphic to a retract of a cell complex, in particular to a CW-complex (def. 14), is a homotopy equivalence (def. 9).

Proof

This is the “Whitehead theorem in model categories”, lemma 10, specialized to Top QuillenTop_{Quillen} via theorem 2.

In proving theorem 2 we have in fact shown a bit more that stated. Looking back, all the structure of Top QuillenTop_{Quillen} is entirely induced by the set I TopI_{Top} (def. 13) of generating cofibrations and the set J TopJ_{Top} (def. 16) of generating acyclic cofibrations (whence the terminology). This phenomenon will keep recurring and will keep being useful as we construct further model categories, such as the classical model structure on pointed topological spaces (def. 44), the projective model structure on topological functors (thm. 5), and finally various model structures on spectra which we turn to in the section on stable homotopy theory.

Therefore we make this situation explicit:

Definition

A model category 𝒞\mathcal{C} (def. 20) is called cofibrantly generated if there exists two subsets

I,JMor(𝒞) I, J \subset Mor(\mathcal{C})

of its class of morphisms, such that

  1. II and JJ have small domains according to def. 25,

  2. the (acyclic) cofibrations of 𝒞\mathcal{C} are precisely the retracts, of II-relative cell complexes (JJ-relative cell complexes), def. 15.

Proposition

For 𝒞\mathcal{C} a cofibrantly generated model category, def. 38, with generating (acylic) cofibrations II (JJ), then its classes W,Fib,CofW, Fib, Cof of weak equivalences, fibrations and cofibrations are equivalently expressed as injective or projective morphisms (def. 21) this way:

  1. Cof=(IInj)ProjCof = (I Inj) Proj

  2. WFib=IInjW \cap Fib = I Inj;

  3. WCof=(JInj)ProjW \cap Cof = (J Inj) Proj;

  4. Fib=JInjFib = J Inj;

Proof

It is clear from the definition that I(IInj)ProjI \subset (I Inj) Proj, so that the closure property of prop. 6 gives an inclusion

Cof(IInj)Proj. Cof \subset (I Inj) Proj \,.

For the converse inclusion, let f(IInj)Projf \in (I Inj) Proj. By the small object argument, prop. 9, there is a factorization f:ICellIInjf\colon \overset{\in I Cell}{\longrightarrow}\overset{I Inj}{\longrightarrow}. Hence by assumption and by the retract argument lemma 7, ff is a retract of an II-relative cell complex, hence is in CofCof.

This proves the first statement. Together with the closure properties of prop. 6, this implies the second claim.

The proof of the third and fourth item is directly analogous, just with JJ replaced for II.

The classical homotopy category

With the classical model structure on topological spaces in hand, we now have good control over the classical homotopy category:

Definition

The Serre-Quillen classical homotopy category is the homotopy category, def. 28, of the classical model structure on topological spaces Top QuillenTop_{Quillen} from theorem 2: we write

Ho(Top)Ho(Top Quillen). Ho(Top) \coloneqq Ho(Top_{Quillen}) \,.
Remark

From just theorem 2, the definition 28 (def. 39) gives that

Ho(Top Quillen)(Top Retract(Cell))/ Ho(Top_{Quillen}) \simeq (Top_{Retract(Cell)})/_\sim

is the category whose objects are retracts of cell complexes (def. 14) and whose morphisms are homotopy classes of continuous functions. But in fact more is true:

Theorem 2 in itself implies that every topological space is weakly equivalent to a retract of a cell complex, def. 14. But by the existence of CW approximations, this cell complex may even be taken to be a CW complex.

(Better yet, there is Quillen equivalence to the classical model structure on simplicial sets which implies a functorial CW approximation |SingX|W clX{\vert Sing X\vert} \overset{\in W_{cl}}{\longrightarrow} X given by forming the geometric realization of the singular simplicial complex of XX.)

Hence the Serre-Quillen classical homotopy category is also equivalently the category of just the CW-complexes whith homotopy classes of continuous functions between them

Ho(Top Quillen) (Top Retract(Cell))/ (Top CW)/ . \begin{aligned} Ho(Top_{Quillen}) & \simeq (Top_{Retract(Cell)})/_\sim \\ & \simeq (Top_{CW})/_{\sim} \end{aligned} \,.

It follows that the universal property of the homotopy category (theorem 1)

Ho(Top Quillen)Top[W cl 1] Ho(Top_{Quillen}) \simeq Top[W_{cl}^{-1}]

implies that there is a bijection, up to natural isomorphism, between

  1. functors out of Top CWTop_{CW} which agree on homotopy-equivalent maps;

  2. functors out of all of TopTop which send weak homotopy equivalences to isomorphisms.

This statement in particular serves to show that two different axiomatizations of generalized (Eilenberg-Steenrod) cohomology theories are equivalent to each other. See at Introduction to Stable homotopy theory -- S the section generalized cohomology functors (this prop.)

Beware that, by remark 4, what is not equivalent to Ho(Top Quillen)Ho(Top_{Quillen}) is the category

hTopTop/ hTop \coloneqq Top/_\sim

obtained from all topological spaces with morphisms the homotopy classes of continuous functions. This category is “too large”, the correct homotopy category is just the genuine full subcategory

Ho(Top Quillen)(Top Retract(Cell))/ Top/ =hTop. Ho(Top_{Quillen}) \simeq (Top_{Retract(Cell)})/_\sim \simeq Top/_\sim = \hookrightarrow hTop \,.

Beware also the ambiguity of terminology: “classical homotopy category” some literature refers to hTophTop instead of Ho(Top Quillen)Ho(Top_{Quillen}). However, here we never have any use for hTophTop and will not mention it again.

Proposition

Let XX be a CW-complex, def. 14. Then the standard topological cylinder of def. 6

XX(i 0,i 1)X×IX X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} X\times I \longrightarrow X

(obtained by forming the product space with the standard topological interval I=[0,1]I = [0,1]) is indeed a cylinder object in the abstract sense of def. 26.

Proof

We describe the proof informally. It is immediate how to turn this into a formal proof, but the notation becomes tedious. (One place where it is spelled out completely is Ottina 14, prop. 2.9.)

So let X 0X 1X 2XX_0 \to X_1 \to X_2\to \cdots \to X be a presentation of XX as a CW-complex. Proceed by induction on the cell dimension.

First observe that the cylinder X 0×IX_0 \times I over X 0X_0 is a cell complex: First X 0X_0 itself is a disjoint union of points. Adding a second copy for every point (i.e. attaching along S 1D 0S^{-1}\to D^0) yields X 0X 0X_0 \sqcup X_0, then attaching an inteval between any two corresponding points (along S 0D 1S^0 \to D^1) yields X 0×IX_0 \times I.

So assume that for nn \in \mathbb{N} it has been shown that X n×IX_n \times I has the structure of a CW-complex of dimension (n+1)(n+1). Then for each cell of X n+1X_{n+1}, attach it twice to X n×IX_n \times I, once at X n×{0}X_n \times \{0\}, and once at X n×{1}X_n \times \{1\}.

The result is X n+1X_{n+1} with a hollow cylinder erected over each of its (n+1)(n+1)-cells. Now fill these hollow cylinders (along S n+1D n+1S^{n+1} \to D^{n+1}) to obtain X n+1×IX_{n+1}\times I.

This completes the induction, hence the proof of the CW-structure on X×IX\times I.

The construction also manifestly exhibits the inclusion XX(i 0,i 1)X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} as a relative cell complex.

Finally, it is clear (prop. 3) that X×IXX \times I \to X is a weak homotopy equivalence.

Conversely:

Proposition

Let XX be any topological space. Then the standard topological path space object (def. 11)

XX I(X δ 0,X δ 1)X×X X \longrightarrow X^I \overset{(X^{\delta_0}, X^{\delta_1})}{\longrightarrow} X \times X

(obtained by forming the mapping space, def. 5, with the standard topological interval I=[0,1]I = [0,1]) is indeed a path space object in the abstract sense of def. 26.

Proof

To see that const:XX Iconst \colon X\to X^I is a weak homotopy equivalence it is sufficient, by prop. 3, to exhibit a homotopy equivalence. Let the homotopy inverse be X δ 0:X IXX^{\delta_0} \colon X^I \to X. Then the composite

XconstX IX δ 0X X \overset{const}{\longrightarrow} X^I \overset{X^{\delta_0}}{\longrightarrow} X

is already equal to the identity. The other we round, the rescaling of paths provides the required homotopy

I×X I (t,γ)γ(t()) X I. \array{ I \times X^I &\overset{(t,\gamma)\mapsto \gamma(t\cdot(-))}{\longrightarrow}& X^I } \,.

To see that X IX×XX^I \to X\times X is a fibration, we need to show that every commuting square of the form

D n X I i 0 D n×I X×X \array{ D^n &\longrightarrow& X^I \\ {}^{\mathllap{i_0}}\downarrow && \downarrow^{} \\ D^n \times I &\longrightarrow& X \times X }

has a lift.

Now first use the adjunction (I×())() I(I \times (-))\dashv (-)^I from prop. 2 to rewrite this equivalently as the following commuting square:

D nD n (i 0,i 0) (D n×I)(D n×I) (i 0,i 1) D n×I X. \array{ D^n \sqcup D^n &\overset{(i_0, i_0)}{\longrightarrow}& (D^n \times I) \sqcup (D^n \times I) \\ {}^{\mathllap{(i_0, i_1)}}\downarrow && \downarrow \\ D^n \times I &\longrightarrow& X } \,.

This square is equivalently (example 9) a morphism out of the pushout

D n×ID nD n((D n×I)(D n×I))X. D^n \times I \underset{D^n \sqcup D^n }{\sqcup} \left((D^n \times I) \sqcup (D^n \times I)\right) \longrightarrow X \,.

By the same reasoning, a lift in the original diagram is now equivalently a lifting in

D n×ID nD n((D n×I)(D n×I)) X (D n×I)×I *. \array{ D^n \times I \underset{D^n \sqcup D^n }{\sqcup} \left((D^n \times I) \sqcup (D^n \times I)\right) &\longrightarrow& X \\ \downarrow && \downarrow \\ (D^n \times I)\times I &\longrightarrow& \ast } \,.

Inspection of the component maps shows that the left vertical morphism here is the inclusion into the square times D nD^n of three of its faces times D nD^n. This is homeomorphic to the inclusion D n+1D n+1×ID^{n+1} \to D^{n+1} \times I (as in remark 7). Therefore a lift in this square exsists, and hence a lift in the original square exists.

Model structure on pointed spaces

A pointed object (X,x)(X,x) is of course an object XX equipped with a point x:*Xx \colon \ast \to X, and a morphism of pointed objects (X,x)(Y,y)(X,x) \longrightarrow (Y,y) is a morphism XYX \longrightarrow Y that takes xx to yy. Trivial as this is in itself, it is good to record some basic facts, which we do here.

Passing to pointed objects is also the first step in linearizing classical homotopy theory to stable homotopy theory. In particular, every category of pointed objects has a zero object, hence has zero morphisms. And crucially, if the original category had Cartesian products, then its pointed objects canonically inherit a non-cartesian tensor product: the smash product. These ingredients will be key below in the section on stable homotopy theory.

Definition

Let 𝒞\mathcal{C} be a category and let X𝒞X \in \mathcal{C} be an object.

The slice category 𝒞 /X\mathcal{C}_{/X} is the category whose

  • objects are morphisms A X\array{A \\ \downarrow \\ X} in 𝒞\mathcal{C};

  • morphisms are commuting triangles A B X\array{ A && \longrightarrow && B \\ & {}_{}\searrow && \swarrow \\ && X} in 𝒞\mathcal{C}.

Dually, the coslice category 𝒞 X/\mathcal{C}^{X/} is the category whose

  • objects are morphisms X A\array{X \\ \downarrow \\ A} in 𝒞\mathcal{C};

  • morphisms are commuting triangles X A B\array{ && X \\ & \swarrow && \searrow \\ A && \longrightarrow && B } in 𝒞\mathcal{C}.

There are the canonical forgetful functors

U:𝒞 /X,𝒞 X/𝒞 U \;\colon \; \mathcal{C}_{/X}, \mathcal{C}^{X/} \longrightarrow \mathcal{C}

given by forgetting the morphisms to/from XX.

We here focus on this class of examples:

Definition

For 𝒞\mathcal{C} a category with terminal object *\ast, the coslice category (def. 40) 𝒞 */\mathcal{C}^{\ast/} is the corresponding category of pointed objects: its

  • objects are morphisms in 𝒞\mathcal{C} of the form *xX\ast \overset{x}{\to} X (hence an object XX equipped with a choice of point; i.e. a pointed object);

  • morphisms are commuting triangles of the form

    * x y X f Y \array{ && \ast \\ & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}} \\ X && \overset{f}{\longrightarrow} && Y }

    (hence morphisms in 𝒞\mathcal{C} which preserve the chosen points).

Remark

In a category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. 41, the terminal object coincides with the initial object, both are given by *𝒞\ast \in \mathcal{C} itself, pointed in the unique way.

In this situation one says that *\ast is a zero object and that 𝒞 */\mathcal{C}^{\ast/} is a pointed category.

It follows that also all hom-sets Hom 𝒞 */(X,Y)Hom_{\mathcal{C}^{\ast/}}(X,Y) of 𝒞 */\mathcal{C}^{\ast/} are canonically pointed sets, pointed by the zero morphism

0:X!0!Y. 0 \;\colon\; X \overset{\exists !}{\longrightarrow} 0 \overset{\exists !}{\longrightarrow} Y \,.
Definition

Let 𝒞\mathcal{C} be a category with terminal object and finite colimits. Then the forgetful functor U:𝒞 */𝒞U \colon \mathcal{C}^{\ast/} \to \mathcal{C} from its category of pointed objects, def. 41, has a left adjoint

𝒞 */U() +𝒞 \mathcal{C}^{\ast/} \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} \mathcal{C}

given by forming the disjoint union (coproduct) with a base point (“adjoining a base point”).

Proposition

Let 𝒞\mathcal{C} be a category with all limits and colimits. Then also the category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. 41, has all limits and colimits.

Moreover:

  1. the limits are the limits of the underlying diagrams in 𝒞\mathcal{C}, with the base point of the limit induced by its universal property in 𝒞\mathcal{C};

  2. the colimits are the limits in 𝒞\mathcal{C} of the diagrams with the basepoint adjoined.

Proof

It is immediate to check the relevant universal property. For details see at slice category – limits and colimits.

Example

Given two pointed objects (X,x)(X,x) and (Y,y)(Y,y), then:

  1. their product in 𝒞 */\mathcal{C}^{\ast/} is simply (X×Y,(x,y))(X\times Y, (x,y));

  2. their coproduct in 𝒞 */\mathcal{C}^{\ast/} has to be computed using the second clause in prop. 23: since the point *\ast has to be adjoined to the diagram, it is given not by the coproduct in 𝒞\mathcal{C}, but by the pushout in 𝒞\mathcal{C} of the form:

    * x X y (po) Y XY. \array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \vee Y } \,.

    This is called the wedge sum operation on pointed objects.

Generally for a set {X i} iI\{X_i\}_{i \in I} in Top */Top^{\ast/}

  1. their product is formed in TopTop as in example 6, with the new basepoint canonically induced;

  2. their coproduct is formed by the colimit in TopTop over the diagram with a basepoint adjoined, and is called the wedge sum iIX i\vee_{i \in I} X_i.

Example

For XX a CW-complex, def. 14 then for every nn \in \mathbb{N} the quotient (example 10) of its nn-skeleton by its (n1)(n-1)-skeleton is the wedge sum, def. 20, of nn-spheres, one for each nn-cell of XX:

X n/X n1iI nS n. X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n \,.
Definition

For 𝒞 */\mathcal{C}^{\ast/} a category of pointed objects with finite limits and finite colimits, the smash product is the functor

()():𝒞 */×𝒞 */𝒞 */ (-)\wedge(-) \;\colon\; \mathcal{C}^{\ast/} \times \mathcal{C}^{\ast/} \longrightarrow \mathcal{C}^{\ast/}

given by

XY*XY(X×Y), X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,,

hence by the pushout in 𝒞\mathcal{C}

XY (id X,y),(x,id Y) X×Y * XY. \array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow && \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,.

In terms of the wedge sum from def. 20, this may be written concisely as

XY=X×YXY. X \wedge Y = \frac{X\times Y}{X \vee Y} \,.
Remark

For a general category 𝒞\mathcal{C} in def. 43, the smash product need not be associative, namely it fails to be associative if the functor ()×Z(-)\times Z does not preserve the quotients involved in the definition.

In particular this may happen for 𝒞=\mathcal{C} = Top.

A sufficient condition for ()×Z(-) \times Z to preserve quotients is that it is a left adjoint functor. This is the case in the smaller subcategory of compactly generated topological spaces, we come to this in prop. 26 below.

These two operations are going to be ubiquituous in stable homotopy theory:

symbolnamecategory theory
XYX \vee Ywedge sumcoproduct in 𝒞 */\mathcal{C}^{\ast/}
XYX \wedge Ysmash producttensor product in 𝒞 */\mathcal{C}^{\ast/}
Example

For X,YTopX, Y \in Top, with X +,Y +Top */X_+,Y_+ \in Top^{\ast/}, def. 42, then

  • X +Y +(XY) +X_+ \vee Y_+ \simeq (X \sqcup Y)_+;

  • X +Y +(X×Y) +X_+ \wedge Y_+ \simeq (X \times Y)_+.

Proof

By example 20, X +Y +X_+ \vee Y_+ is given by the colimit in TopTop over the diagram

* X * * Y. \array{ && && \ast \\ && & \swarrow && \searrow \\ X &\,\,& \ast && && \ast &\,\,& Y } \,.

This is clearly X*YX \sqcup \ast \sqcup Y. Then, by definition 43

X +Y + (X*)×(X*)(X*)(Y*) X×YXY*XY* X×Y*. \begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned}
Example

Let 𝒞 */=Top */\mathcal{C}^{\ast/} = Top^{\ast/} be pointed topological spaces. Then

I +Top */ I_+ \in Top^{\ast/}

denotes the standard interval object I=[0,1]I = [0,1] from def. 6, with a djoint basepoint adjoined, def. 42. Now for XX any pointed topological space, then

X(I +)=(X×I)/({x 0}×I) X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)

is the reduced cylinder over XX: the result of forming the ordinary cyclinder over XX as in def. 6, and then identifying the interval over the basepoint of XX with the point.

(Generally, any construction in 𝒞\mathcal{C} properly adapted to pointed objects 𝒞 */\mathcal{C}^{\ast/} is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)

Just like the ordinary cylinder X×IX\times I receives a canonical injection from the coproduct XXX \sqcup X formed in TopTop, so the reduced cyclinder receives a canonical injection from the coproduct XXX \sqcup X formed in Top */Top^{\ast/}, which is the wedge sum from example 20:

XXX(I +). X \vee X \longrightarrow X \wedge (I_+) \,.
Example

For (X,x),(Y,y)(X,x),(Y,y) pointed topological spaces with YY a locally compact topological space, then the pointed mapping space is the topological subspace of the mapping space of def. 5

Maps((Y,y),(X,x)) *(X Y,const x) Maps((Y,y),(X,x))_\ast \hookrightarrow (X^Y, const_x)

on those maps which preserve the basepoints, and pointed by the map constant on the basepoint of XX.

In particular, the standard topological pointed path space object on some pointed XX (the pointed variant of def. 11) is the pointed mapping space Maps(I +,X) *Maps(I_+,X)_\ast.

The pointed consequence of prop. 2 then gives that there is a natural bijection

Hom Top */((Z,z)(Y,y),(X,x))Hom Top */((Z,z),Maps((Y,y),(X,x)) *) Hom_{Top^{\ast/}}((Z,z) \wedge (Y,y), (X,x)) \simeq Hom_{Top^{\ast/}}((Z,z), Maps((Y,y),(X,x))_\ast)

between basepoint-preserving continuous functions out of a smash product, def. 43, with pointed continuous functions of one variable into the pointed mapping space.

Example

Given a morphism f:XYf \colon X \longrightarrow Y in a category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. 41, with finite limits and colimits,

  1. its fiber or kernel is the pullback of the point inclusion

    fib(f) X (pb) f * Y \array{ fib(f) &\longrightarrow& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y }
  2. its cofiber or cokernel is the pushout of the point projection

    X f Y (po) * cofib(f). \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& cofib(f) } \,.
Remark

In the situation of example 25, both the pullback as well as the pushout are equivalently computed in 𝒞\mathcal{C}. For the pullback this is the first clause of prop. 23. The second clause says that for computing the pushout in 𝒞\mathcal{C}, first the point is to be adjoined to the diagram, and then the colimit over the larger diagram

* X f Y * \array{ \ast \\ & \searrow \\ & & X &\overset{f}{\longrightarrow}& Y \\ & & \downarrow && \\ & & \ast && }

be computed. But one readily checks that in this special case this does not affect the result. (The technical jargon is that the inclusion of the smaller diagram into the larger one in this case happens to be a final functor.)

Proposition

Let 𝒞\mathcal{C} be a model category and let X𝒞X \in \mathcal{C} be an object. Then both the slice category 𝒞 /X\mathcal{C}_{/X} as well as the coslice category 𝒞 X/\mathcal{C}^{X/}, def. 40, carry model structures themselves – the model structure on a (co-)slice category, where a morphism is a weak equivalence, fibration or cofibration iff its image under the forgetful functor UU is so in 𝒞\mathcal{C}.

In particular the category 𝒞 */\mathcal{C}^{\ast/} of pointed objects, def. 41, in a model category 𝒞\mathcal{C} becomes itself a model category this way.

The corresponding homotopy category of a model category, def. 28, we call the pointed homotopy category Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}).

Proof

This is immediate:

By prop. 23 the (co-)slice category has all limits and colimits. By definition of the weak equivalences in the (co-)slice, they satisfy two-out-of-three, def. 19, because the do in 𝒞\mathcal{C}.

Similarly, the factorization and lifting is all induced by 𝒞\mathcal{C}: Consider the coslice category 𝒞 X/\mathcal{C}^{X/}, the case of the slice category is formally dual; then if

X A f B \array{ && X \\ & \swarrow && \searrow \\ A && \underset{f}{\longrightarrow} && B }

commutes in 𝒞\mathcal{C}, and a factorization of ff exists in 𝒞\mathcal{C}, it uniquely makes this diagram commute

X A C B. \array{ && X \\ & \swarrow &\downarrow& \searrow \\ A &\longrightarrow& C & \longrightarrow& B } \,.

Similarly, if

A C B D \array{ A &\longrightarrow& C \\ \downarrow && \downarrow \\ B &\longrightarrow& D }

is a commuting diagram in 𝒞 X/\mathcal{C}^{X/}, hence a commuting diagram in 𝒞\mathcal{C} as shown, with all objects equipped with compatible morphisms from XX, then inspection shows that any lift in the diagram necessarily respects the maps from XX, too.

Example

For 𝒞\mathcal{C} any model category, with 𝒞 */\mathcal{C}^{\ast/} its pointed model structure according to prop. 24, then the corresponding homotopy category (def. 28) is, by remark 17, canonically enriched in pointed sets, in that its hom-functor is of the form

[,] *:Ho(𝒞 */) op×Ho(𝒞 */)Set */. [-,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/})^\op \times Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/} \,.
Definition

Write Top Quillen */Top^{\ast/}_{Quillen} for the classical model structure on pointed topological spaces, obtained from the classical model structure on topological spaces Top QuillenTop_{Quillen} (theorem 2) via the induced coslice model structure of prop. 24.

Its homotopy category, def. 28,

Ho(Top */)Ho(Top Quillen */) Ho(Top^{\ast/}) \coloneqq Ho(Top_{Quillen}^{\ast/})

we call the classical pointed homotopy category.

Remark

The fibrant objects in the pointed model structure 𝒞 */\mathcal{C}^{\ast/}, prop. 24, are those that are fibrant as objects of 𝒞\mathcal{C}. But the cofibrant objects in 𝒞 *\mathcal{C}^{\ast} are now those for which the basepoint inclusion is a cofibration in XX.

For 𝒞 */=Top Quillen */\mathcal{C}^{\ast/} = Top^{\ast/}_{Quillen} from def. 44, then the corresponding cofibrant pointed topological spaces are tyically referred to as spaces with non-degenerate basepoints or . Notice that the point itself is cofibrant in Top QuillenTop_{Quillen}, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.

While the existence of the model structure on Top */Top^{\ast/} is immediate, via prop. 24, for the discussion of topologically enriched functors (below) it is useful to record that this, too, is a cofibrantly generated model category (def. 38), as follows:

Definition

Write

I Top */={S + n1(ι n) +D + n}Mor(Top */) I_{Top^{\ast/}} = \left\{ S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+ \right\} \;\; \subset Mor(Top^{\ast/})

and

J Top */={D + n(id,δ 0) +(D n×I) +}Mor(Top */), J_{Top^{\ast/}} = \left\{ D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+ \right\} \;\;\; \subset Mor(Top^{\ast/}) \,,

respectively, for the sets of morphisms obtained from the classical generating cofibrations, def. 13, and the classical generating acyclic cofibrations, def. 16, under adjoining of basepoints (def. 42).

Theorem

The sets I Top */I_{Top^{\ast/}} and J Top */J_{Top^{\ast/}} in def. 45 exhibit the classical model structure on pointed topological spaces Top Quillen */Top^{\ast/}_{Quillen} of def. 44 as a cofibrantly generated model category, def. 38.

(This is also a special case of a general statement about cofibrant generation of coslice model structures, see this proposition.)

Proof

Due to the fact that in J Top */J_{Top^{\ast/}} a basepoint is freely adjoined, lemma 5 goes through verbatim for the pointed case, with J TopJ_{Top} replaced by J Top */J_{Top^{\ast/}}, as do the other two lemmas above that depend on point-set topology, lemma 1 and lemma 4. With this, the rest of the proof follows by the same general abstract reasoning as above in the proof of theorem 2.

Model structure on compactly generated spaces

The category Top has the technical inconvenience that mapping spaces X YX^Y (def. 5) satisfying the exponential property (prop. 2) exist in general only for YY a locally compact topological space, but fail to exist more generally. In other words: Top is not cartesian closed. But cartesian closure is necessary for some purposes of homotopy theory, for instance it ensures that

  1. the smash product (def. 43) on pointed topological spaces is associative (prop. 26 below);

  2. there is a concept of topologically enriched functors with values in topological spaces, to which we turn below;

  3. geometric realization of simplicial sets preserves products.

The first two of these are crucial for the development of stable homotopy theory in the next section, the third is a great convenience in computations.

Now, since the homotopy theory of topological spaces only cares about the CW approximation to any topological space (remark 16), it is plausible to ask for a full subcategory of Top which still contains all CW-complexes, still has all limits and colimits, still supports a model category structure constructed in the same way as above, but which in addition is cartesian closed, and preferably such that the model structure interacts well with the cartesian closure.

Such a full subcategory exists, the category of compactly generated topological spaces. This we briefly describe now.

Literature (Strickland 09)

\,

Definition

Let XX be a topological space.

A subset AXA \subset X is called compactly closed (or kk-closed) if for every continuous function f:KXf \colon K \longrightarrow X out of a compact Hausdorff space KK