用户: Scavenger/Infinity Category and Higher Algebra/Model Categories

The goal of this notes is to collect the basic theory of model categories, with an eye toward its application in the theory of $\infty$-categories. We will mainly follow the definitions and conventions of [1].

Definition 0.1. Let $\varkappa$ be a strongly inaccessible cardinal. A $\varkappa$-model category (or simply a model category) is a category $C$ together with three sets of morphisms in $\mathcal{C}$, called cofibrations, fibrations and weak equivalences respectively (we will call a map a trivial (co)fibration if it is both a (co)fibration and a weak equivalence), such that the following conditions are satisfied:

(i) $\mathcal{C}$ admits $\varkappa$-small limits and colimits.

(ii) Let $f:X\to Y$ and $g:Y\to Z$ be morphisms in $\mathcal{C}$. If any two of $f,g,g\circ f$ are weak equivalences, then so is the third.

(iii) Suppose $f:X\to Y$ is a retract of $g:X^{\prime }\to Y^{\prime }$, then if $g$ is a cofibration (or fibration, or weak equivalence), then so is $f$.

(iv) given a commutative diagram of solid arrows in $\mathcal{C}$:

a dashed arrow can be found rendering the diagram commutative if either $i$ is a cofibration and $p$ is a trivial fibration, or $i$ is a trivial cofibration and $p$ is a fibration.

(v) Any morphism $X\to Z$ in $\mathcal{C}$ admits factorizations

where $f$ is a cofibration, $g$ is a trivial fibration, $f^{\prime }$ is a trivial cofibration and $g^{\prime }$ is a fibration.

From now on we fix a strongly inaccessible cardinal $\varkappa$ as our size parameter unless otherwise stated.

Example 0.2 (The Joyal Model Structure). Let $\mathrm{sSet}$ be the category of $\varkappa$-simplicial sets. We define the Joyal model structure on $\mathrm{sSet}$ as follows:

•  The cofibrations are the monomorphisms.

•  The fibrations are the $\varkappa$-isofibrations.

•  The weak equivalences are the categorical equivalences.

Example 0.3. Let $\mathcal{C}$ be a model category, and let $X$ be an object of $\mathcal{C}$. The slice category ${\mathcal{C}}_{/X}$ admits a natural model structure defined as follows:

•  A morphism in ${\mathcal{C}}_{/X}$ is a fibration/cofibration/weak equivalence if and only if its underlying morphism in $\mathcal{C}$ is a fibration/cofibration/weak equivalence.

Notation 0.4. By definition, every model category $\mathcal{C}$ has an initial object $\varnothing$ and a final object $\ast$. An object $X$ of $\mathcal{C}$ is called fibrant if the unique map $X\to \ast$ is a fibration and is called cofibrant if the unique map $\varnothing \to X$ is a cofibration. An object is called fibrant-cofibrant if it is both fibrant and cofibrant.

The Homotopy Category of a Model Category

We turn to define the homotopy category of a model category.

Let $\mathcal{C}$ be a model category, let $X$ be a cofibrant object of $C$ and let $Y$ be a fibrant object. There is an equivalence relation on the set ${\mathrm{Mor}}_{\mathcal{C}}(X,Y)$ called being “homotopic” (for a detailed definition, see [1] Subsection A.2.2).

Definition 0.5. Let $\mathcal{C}$ be a model category. We define a new category $h\mathcal{C}$, called the homotopy category of $\mathcal{C}$, as follows:

•  The objects of $h\mathcal{C}$ are the fibrant-cofibrant objects of $\mathcal{C}$.

•  Let $X,Y$ be fibrant-cofibrant objects of $\mathcal{C}$, the set ${\mathrm{Mor}}_{h\mathcal{C}}(X,Y)$ is defined to be the quotient of ${\mathrm{Mor}}_{\mathcal{C}}(X,Y)$ by the homotopic equivalence relation discussed above.

•  Composition law and identity morphisms of $h\mathcal{C}$ are given by those of $\mathcal{C}$ (This can be checked to be well defined).

0.6. Let $\mathcal{C}$ be a model category. Let $H(\mathcal{C})$ be the (strict) localization of $\mathcal{C}$ with respect to the set of weak equivalences. Let $X$ be a cofibrant object of $\mathcal{C}$ and let $Y$ be a fibrant object. It can be checked that the functor $\mathcal{C}\to H(\mathcal{C})$ brings homotopic morphisms between $X$ and $Y$ to identical morphisms, and thus induces a functor $h\mathcal{C}\to H(\mathcal{C})$. It can be proved that this functor is an equivalence.

Quillen Adjunctions

Definition 0.7. Let $\mathcal{C}$ and $\mathcal{D}$ be model categories, let $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ be a pair of adjoint functors (such that $F$ is left adjoint to $G$). The following conditions are equivalent:

(i) $F$ preserves cofibrations and trivial cofibrations.

(ii) $G$ preserves fibrations and trivial fibrations.

(iii) $F$ preserves cofibrations and $G$ preserves fibrations.

(iv) $F$ preserves trivial cofibrations and $G$ preserves trivial fibraions.

If these conditions are satisfied, then we say that the pair (F,G) is a Quillen adjunction between $\mathcal{C}$ and $\mathcal{D}$. We also say that $F$ is a left Quillen functor and $G$ is a right Quillen functor. In this case, one can show that $F$ preserves weak equivalences between cofibrant objects and $G$ preserves weak equivalences between fibrant objects.

References

[1]	Jacob Lurie, “Higher Topos Theory”, Available for download at https://www.math.ias.edu/ lurie/