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Definitions
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fundamental (?,1)-category?
Basic facts
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A model category structure $(\mathcal{C}(W,Fib,Cof))$ on some category $\mathcal{C}$ is a means to guarantee the local smallness and to improve the tractability of the homotopy category of the underlying category with weak equivalences $(\mathcal{C},W)$. In particular, if a category with weak equivalences admits a model category structure, then its homotopy category (in the sense of localization $\mathcal{C}[W^{-1}]$ at the class of weak equivalences) is equivalent to the category whose objects are those objects of $\mathcal{C}$ which are both fibrant cofibrant, and whose morphisms are the actual homotopy classes of morphisms between these objects (left homotopy or right homotopy equivalence classes in the sense of homotopy in a model category) .
Let $\mathcal{C}$ be a model category. Write $Ho(\mathcal{C})$ for the category whose
objects are those objects of $\mathcal{C}$ which are both fibrant and cofibrant;
morphisms are the homotopy classes of morphisms of $\mathcal{C}$, hence the equivalence classes of morphism under left homotopy.
This is, up to equivalence of categories, the homotopy category of the model category $\mathcal{C}$.
We spell out that def. 1 indeed satisfies the universal property that defines the homotopy category of a category with weak equivalences.
(Whitehead theorem in model categories)
Let $\mathcal{C}$ be a model category. A weak equivalence between two objects which are both fibrant and cofibrant is a homotopy equivalence.
(e.g. Goerss-Jardine 99, part I, theorem 1.10)
By the factorization axioms in $\mathcal{C}$, every weak equivalence $f\colon X \longrightarrow Y$ factors through an object $Z$ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with $X$ and $Y$ both fibrant and cofibrant, so is $Z$, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.
So let $f \colon X \longrightarrow Y$ be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is formally dual. Then in fact it has a genuine right inverse given by a lift $f^{-1}$ in the diagram
To see that $f^{-1}$ is also a left inverse up to left homotopy, let $Cyl(X)$ be any cylinder object on $X$, hence a factorization of the codiagonal on $X$ as a cofibration followed by a an acyclic fibration
and consider the square
which commutes due to $f^{-1}$ being a genuine right inverse of $f$. By construction, this commuting square now admits a lift $\eta$, and that constitutes a left homotopy $\eta \colon f^{-1}\circ f \Rightarrow_L id$.
Given a model category $\mathcal{C}$, consider a choice for each object $X \in \mathcal{C}$ of
a factorization $\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X$ of the initial morphism, such that when $X$ is already cofibrant then $p_X = id_X$;
a factorization $X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast$ of the terminal morphism, such that when $X$ is already fibrant then $j_X = id_X$.
Write then
for the functor to the homotopy category, def. 1, which sends an object $X$ to the object $R Q X$ and sends a morphism $f \colon X \longrightarrow Y$ to the homotopy class of the result of first lifting in
and then lifting (here: extending) in
The construction in def. 2 is indeed well defined.
First of all, the object $R Q X$ is indeed both fibrant and cofibrant (as well as related by a zig-zag of weak equivalences to $X$):
Now to see that the image on morphisms is well defined. First observe that any two choices $(Q f)_{i}$ of the first lift in the definition are left homotopic to each other, exhibited by lifting in
Hence also the composites $j_{Q Y}\circ (Q_f)_i$ are left homotopic to each other, and since their domain is cofibrant, they are also right homotopic (via this lemma) by a right homotopy $\kappa$. This implies finally, by lifting in
that also $R (Q f)_1$ and $R (Q f)_2$ are right homotopic, hence that indeed $R Q f$ represents a well-defined homotopy class.
Finally to see that the assignment is indeed functorial, observe that the commutativity of the lifting diagrams for $Q f$ and $R Q f$ imply that also the following diagram commutes
Now from the pasting composite
one sees that $(R Q g)\circ (R Q f)$ is a lift of $g \circ f$ and hence the same argument as above gives that it is homotopic to the chosen $R Q(g \circ f)$.
For $\mathcal{C}$ a category with weak equivalences, its homotopy category (or: localization at the weak equivalences) is, if it exists, a category $Ho(\mathcal{C})$ equipped with a functor
which sends weak equivalences to isomorphisms, and which is universal with this property:
for $F \colon \mathcal{C} \longrightarrow D$ any functor out of $\mathcal{C}$ into any category, such that $F$ takes weak equivalences to isomorphisms, it factors through $\gamma$ up to a natural isomorphism
and this factorization is unique up to unique isomorphism, in that for $(\tilde F_1, \rho_1)$ and $(\tilde F_2, \rho_2)$ two such factorizations, then there is a unique natural isomorphism $\kappa \colon \tilde F_1 \Rightarrow \tilde F_2$ making the evident diagram of natural isomorphisms commute.
For $\mathcal{C}$ a model category, the functor $\gamma_{R,Q}$ in def. 2 (for any choice of $R$ and $Q$) exhibits $Ho(\mathcal{C})$ as indeed being the homotopy category of the underlying category with weak equivalences, in the sense of def. 3.
First, to see that that $\gamma$ indeed takes weak equivalences to isomorphisms: By two-out-of-three applied to the commuting diagrams shown in the proof of lemma 2 the morphism $R Q f$ is a weak equivalence if $f$ is:
With this the “Whitehead theorem for model categories”, lemma 1, implies that $R Q f$ represents an isomorphism in $Ho(\mathcal{C})$.
Now let $F \colon \mathcal{C}\longrightarrow D$ be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as
uniquely up to unique natural isomorphism. Now by construction of $R$ and $Q$ in def. 2, $\gamma_{R,Q}$ is the identity on the full subcategory of fibrant-cofibrant objects. It follows that if $\tilde F$ exists at all, it must satisfy for all $X \stackrel{f}{\to} Y$ with $X$ and $Y$ both fibrant and cofibrant that
But by def. 1 that already fixes $\tilde F$ on all of $Ho(\mathcal{C})$, up to unique natural isomorphism. Hence it only remains to check that with this definition of $\tilde F$ there exists any natural isomorphism $\rho$ filling the diagram above.
To that end, apply $F$ to the above commuting diagram to obtain
Here now all horizontal morphisms are isomorphisms, by assumption on $F$. It follows that defining $\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1}$ makes the required natural isomorphism:
Due to theorem 1 we may suppress the choices of cofibrant $Q$ and fibrant replacement $R$ in def. 2 and just speak of the localization functor
up to natural isomorphism.
While the construction of the homotopy category in def. 1 combines the restriction to good (fibrant/cofibrant) objects with the passage to homotopy classes of morphisms, it is often useful to consider intermediate stages:
Given a model category $\mathcal{C}$, write
for the system of full subcategory inclusions on the cofibrant objects ($\mathcal{C}_c$), the fibrant objects ($\mathcal{C}_f$) and the objects which are both fibrant and cofibrant ($\mathcal{C}_{fc}$), all regarded a categories with weak equivalences, via the weak equivalences inherited from $\mathcal{C}$.
Of course the subcategories in def. 4 inherit more structure than just that of categories with weak equivalences from $\mathcal{C}$. $\mathcal{C}_f$ and $\mathcal{C}_c$ each inherit “half” of the factorization axioms. One says that $\mathcal{C}_f$ has the structure of a “fibration category” called a “category of fibrant objects”, while $\mathcal{C}_c$ has the structure of a “cofibration category”.
The proof of theorem 1 immediately implies the following:
For $\mathcal{C}$ a model category, the restriction of the localization functor $\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$ from def. 2 (using remark 1) to any of the sub-categories with weak equivalences of def. 4
exhibits $Ho(\mathcal{C})$ equivalently as the homotopy category also of these subcategories. In particular there are equivalences of categories
For $\mathcal{C}$ and $\mathcal{D}$ two categories with weak equivalences, then a functor $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ is called homotopical functor if it sends weak equivalences to weak equivalences.
Given a homotopical functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ (def. 5) between categories with weak equivalences whose homotopy categories $Ho(\mathcal{C})$ and $Ho(\mathcal{D})$ exist (def. 3), then its derived functor is the functor $Ho(F)$ between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. 3):
While many functors of interest between model categories are not homotopical in the sense of def. 5, many become homotopical after restriction to the full subcategories of fibrant object or of cofibrant objects, def. 4. By corollary 1 this is just as good for the purpose of homotopy theory.
Therefore one considers the following generalization of def. 6:
Consider a functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ out of a model category $\mathcal{C}$ into a category with weak equivalences $\mathcal{D}$.
If the restriction of $F$ to the full subcategory $\mathcal{C}_f$ of fibrant object becomes a homotopical functor (def. 5), then the derived functor of that restriction, according to def. 6, is called the right derived functor of $F$ and denoted by $\mathbb{R}F$:
Here the commuting square on the left is from corollary 1, the square on the right is that of def. 6.
If the restriction of $F$ to the full subcategory $\mathcal{C}_c$ of cofibrant object becomes a homotopical functor (def. 5), then the derived functor of that restriction, according to def. 6, is called the left derived functor of $F$ and denoted by $\mathbb{L}F$:
Here the commuting square on the left is from corollary 1, the square on the right is that of def. 6.
The key fact that makes def. 7 practically useful is the following
Let $\mathcal{C}$ be a model category with full subcategories $\mathcal{C}_f, \mathcal{C}_c$ of fibrant objects and of cofibrant objects respectively (def. 4). Let $\mathcal{D}$ be a category with weak equivalences.
A functor
is a homotopical functor, def. 5, already if it sends acylic fibrations to weak equivalences.
A functor
is a homotopical functor, def. 5, already if it sends acylic cofibrations to weak equivalences.
Let $\mathcal{C}, \mathcal{D}$ be model categories and consider $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ a functor. Then:
If $F$ preserves cofibrant objects and acyclic cofibrations between these, then its left derived functor (def. 7) $\mathbb{L}F$ exists, fitting into a diagram
If $F$ preserves fibrant objects and acyclic fibrants between these, then its right derived functor (def. 7) $\mathbb{R}F$ exists, fitting into a diagram
In practice it turns out to be useful to arrange for the assumptions in corollary 2 to be satisfied in the following neat way:
Let $\mathcal{C}, \mathcal{D}$ be model categories. A pair of adjoint functors between them
is called a Quillen adjunction (and $L$,$R$ are called left/right Quillen functors, respectively) if the following equivalent conditions are satisfied
$L$ preserves cofibrations and $R$ preserves fibrations;
$L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations;
$L$ preserves cofibrations and acylic cofibrations;
$R$ preserves fibrations and acyclic fibrations.
The conditions in def. 8 are indeed all equivalent.
Observe that
(i) A left adjoint $L$ between model categories preserves acyclic cofibrations precisely if its right adjoint $R$ preserves fibrations.
(ii) A left adjoint $L$ between model categories preserves cofibrations precisely if its right adjoint $R$ preserves acyclic fibrations.
We discuss statement (i), statement (ii) is formally dual. So let $f\colon A \to B$ be an acyclic cofibration in $\mathcal{D}$ and $g \colon X \to Y$ a fibration in $\mathcal{C}$. Then for every commuting diagram as on the left of the following, its $(L\dashv R)$-adjunct is a commuting diagram as on the right here:
If $L$ preserves acyclic cofibrations, then the diagram on the right has a lift, and so the $(L\dashv R)$-adjunct of that lift is a lift of the left diagram. This shows that $R(g)$ has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if $R$ preserves fibrations, the same argument run from right to left gives that $L$ preserves acyclic fibrations.
Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.
For $\mathcal{C}, \mathcal{D}$ two model categories, a Quillen adjunction (def.8)
is called a Quillen equivalence if the following equivalent conditions hold.
The right derived functor of $R$ (via prop. 2, corollary 2) is an equivalence of categories
The left derived functor of $L$ (via prop. 2, corollary 2) is an equivalence of categories
For every cofibrant object $d \in \mathcal{D}$ and every fibrant object $c \in \mathcal{C}$, a morphism $d \longrightarrow R(c)$ is a weak equivalence precisely if its adjunct morphism $L(c) \to d$ is
For every cofibrant object $d\in \mathcal{C}$, the composite
(of the adjunction unit with any fibrant replacement) is a weak equivalence.
For every fibrant object $c \in \mathcal{C}$, the composite
(of the adjunction counit with any cofibrant replacement) is a weak equivalence in $D$.
The original account is due to
Review includes
William Dwyer, J. Spalinski, Homotopy theories and model categories (pdf) in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology 1995
Paul Goerss, Rick Jardine, section II.1 of Simplicial homotopy theory Birkhäuser 1999, 2009