Lecture 1

1.1 Model Categories

Definition 1.1.1. Given a commutative square of the following form:

$$\label{liftingdiagram} \begin{split} \xymatrix{A \ar[r]^ f \ar[d]_ i& X \ar[d]^ p \\ B \ar[r]_ g & Y} \end{split}$$

a lift in the diagram is a morphism $h \colon B \to X$ such that $hi=f$ and $ph=g$. A morphism $i \colon A \to B$ is said to have the left lifting property (LLP) with respect to another morphism $p \colon X \to Y$ and $p$ is said to have the right lifting property (RLP) with respect to $i$ if a lifting exists for any choice of $f$ and $g$ making the above diagram commute.

Definition 1.1.2. Let $f \colon X \to Y$ be a morphism in a category $\mathcal{C}$, then a retraction of $f$ is left-inverse. That is, there exists a morphism $r$ such that $r \circ f = \mathrm{id}_ X$.

Definition 1.1.3. A model category is a category $\mathcal{C}$ with three distinguished classes of morphisms:

• Weak equivalences - $\mathrm{W}_\mathcal {C}$;
• Fibrations - $\mathrm{Fib}_\mathcal {C}$;
• Cofibrations - $\mathrm{Cof}_\mathcal {C}$;
each of which is closed under composition. We say that a morphism which is both a fibration and a weak equivalence is an acyclic fibration, and dually a morphism which is both a cofibration and a weak equivalence is an acyclic cofibration. The distinguished classes of morphisms and the category $\mathcal{C}$ must satisfy the following axioms:

• (MC1) $\mathcal{C}$ has all small limits and colimits. In particular, there is an initial object $\emptyset$ and a terminal object $\ast$.
• (MC2) If $f$ and $g$ are morphism such that $gf$ is defined and if two of $f$, $g$ and $gf$ are weak equivalences, then so is the third. That is, the weak equivalences satisfy the 2-out-of-3 property.
• (MC3) The three distinguished classes of morphisms are closed under retracts.
• (MC4) Given a commutative diagram of the form above, a lift exists when either $i$ is a cofibration and $p$ is an acyclic fibration or when $i$ is an acyclic cofibration and $p$ is a fibration.
• (MC5) Each morphism $f$ in $\mathcal{C}$ can be factored in two ways:
1. $f = pi$, where $i$ is a cofibration and $p$ is an acyclic fibration.
2. $f = pi$, where $p$ is a fibration and $i$ is an acyclic cofibration.

Remark 1.1.4. There are several important things that should be highlighted in regards to the definition of a model category.

1. We shall assume throughout that our category has all small limits and colimits. It is, in fact, enough to just assume finite limits and colimits, and indeed this is the original definition of Quillen. However, the proofs that we will refer to for many result are taken from the book of Hovey [Hov99], in which the existence of all small limits and colimits is assumed.
2. We assume, unless stated otherwise, that our model structures admit functorial factorizations. This allows us to pick the factorizations of axiom MC5 in a functorial way. Again, this is one of the assumptions made by Hovey [Hov99], motived by work of Thomason [Wei01] and Dwyer-Hirschhorn-Kan-Smith [DHKS04] which makes various constructions more manageable. We refer to [Definition 12.1.1, Rie14] or [Definition 3.5, BR13] for a definition of what it means to have functorial factorization, which is akin to asking for a section of the functor $(d^1)^\ast \colon \mathcal{C}^{[3]} \to \mathcal{C}^{[2]}$ where $d^1 \colon [2] \to [3]$ is such that $1 \mapsto 1$ and $2 \mapsto 3$. Note that the definition of functorial factorization in [Definition 1.1.1.2, Hov99] is not complete as discussed in [MSE].
3. The data of weak equivalences, fibrations and cofibrations in a category $\mathcal{C}$ satisfying (MC2)-(MC5) is said to be the data of a model structure on $\mathcal{C}$. A category $\mathcal{C}$ may support a model structure while not actually forming a model category (e.g., if $\mathcal{C}$ fails to have a terminal object).
4. Finally, we note that in many sources, including [Hov99], one will find the word trivial used instead of acyclic. We have chosen to work with the latter as a personal preference.

Definition 1.1.5. Let $\mathcal{C}$ be a model category. An object $X \in \mathcal{C}$ is said to be:

• fibrant if the unique morphism $X \to \ast$ is a fibration;
• cofibrant is the unique morphism $\emptyset \to X$ is a cofibration;
• bifibrant if it is both fibrant and cofibrant.

Using the factorizations of morphisms given in MC5 it is possible to find fibrant and cofibrant replacements for any object in a model category by factoring the unique morphisms $X \to \ast$ and $\emptyset \to X$ respectively.

Definition 1.1.6. Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$.

• A fibrant replacement of $X$ is an object $RX$ equipped with an acyclic cofibration $X \to RX$ such that $RX$ is fibrant.
• A cofibrant replacement of $X$ is an object $QX$ along with an acyclic fibration $QX \to X$ such that $QX$ is cofibrant.

Note that although many such replacements may exist, we are given one such replacement that is functorial by assumption.

Remark 1.1.7. We highlight that the fibrant replacement of a cofibrant object, that is, $RQX$ is still a cofibrant object, and dually the object $QRX$ is still fibrant. As such there is a notion of a bifibrant replacement as the composite, and there is a weak equivalence $RQX \xrightarrow {\sim } QRX$ [3.3, Rie19].

Proposition 1.1.8 ([Corollary 1.2, Qui69]). The class of cofibrations are the morphisms which have the LLP with respect to all acyclic fibrations. Dually, the fibrations are the morphisms which have the RLP with respect to all acyclic cofibrations.

Corollary 1.1.9. The class of fibrations (resp., cofibrations) is closed under pullbacks (resp., pushouts).

Corollary 1.1.10. A morphism $f \colon X \to Y$ is in all three classes of distinguished morphisms if and only if it is an isomorphism.

The above proposition tells us that after the weak equivalences have been defined, we can use the fibrations (resp., cofibrations) to construct the cofibrations (resp., fibrations) via liftings. That is, the weak equivalences along with one of the other classes of morphisms is enough to uniquely determine a model structure.

There is an enlightening discussion on MathOverflow entitled "What determines a model structure?" which explores other collections of incomplete data which are enough to uniquely determine a model structure. We summarize some of the results from this discussion here. The proof of the following result uses the useful fact that any weak equivalence can be written as a composite of an acyclic cofibration and an acyclic fibration using the factorization axiom.

Proposition 1.1.11 ([Proposition 1.37, Joy]). A model structure is uniquely determined by the data of:

• Cofibrations and fibrant objects.
• Fibrations and cofibrant objects.
• Fibrations and cofibrations.
• Acyclic fibrations and acyclic cofibrations.

Conversely, the following data does not uniquely describe a model structure:

• Weak equivalences.
• Cofibrant objects and weak equivalences.
• Fibrant objects and weak equivalences.
• Fibrant objects and cofibrant objects.

1.2 The Homotopy Category

The homotopy category is the category that one obtains by formally inverting the weak equivalences in a model category. This allows us to then consider objects up to some notion of homotopy, akin to the classical cases of topological spaces.

In particular, given a model category $\mathcal{C}$ the homotopy category will be a (1-)category $\operatorname {Ho}(\mathcal{C})$ equipped with a functor $\gamma \colon \mathcal{C} \to \operatorname {Ho}(\mathcal{C})$ which is universal among functors which sends $\operatorname {W}_\mathcal {C}$ to isomorphisms. The highly structured nature of model categories allows this process to be done in a controlled manner. First we need to discuss the notion of homotopy in model categories

Definition 1.2.1. Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$. A cylinder object $\operatorname {Cyl}(X)$ for $X$ is a factorization of the codiagonal $\nabla _ X \colon X \sqcup X \to X$ as

$\nabla _ X \colon X \sqcup X \xrightarrow {(i_0,i_1) \in \operatorname {Cof}_\mathcal {C}} \operatorname {Cyl}(X) \xrightarrow {p \in \operatorname {W}_\mathcal {C}} X.$

Note that such a factorization exists by MC5.

Definition 1.2.2. Let $f,g \colon X \to Y$ be a pair of morphisms in a model category. Then a left homotopy $\eta \colon f \sim _ L g$ is a morphism $\eta \colon \operatorname {Cyl}(X) \to Y$ which makes the following diagram commute:

$\xymatrix{ X \ar[dr]_f \ar[r]^-{i_0} & \operatorname{Cyl}(X) \ar[d]^\eta \ar@{<-}[r]^-{i_1} & X \ar[dl]^g \\ & Y }$

We can also consider a dual notion of path objects constructed out of fibration data.

Definition 1.2.3. Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$. A path object $\operatorname {Path}(X)$ for $X$ is a factorization of the diagonal $\Delta _ X \colon X\to X \times X$ as

$\Delta _ X \colon X \xrightarrow {s \in \operatorname {W}_\mathcal {C}} \operatorname {Path}(X) \xrightarrow {(d_0,d_1) \in \operatorname {Fib}_\mathcal {C}} X \times X .$

Note that such a factorization exists by MC5

Definition 1.2.4. Let $f,g \colon X \to Y$ be a pair of morphisms in a model category. Then a right homotopy $\eta \colon f \sim _ R g$ is a morphism $\eta \colon X \to \operatorname {Path}(Y)$ which makes the following diagram commute:

$\xymatrix{ & X \ar[d]^{\eta } \ar[dl]_{f} \ar[dr]^{g} \\ Y & \operatorname {Path}(Y) \ar[r]_-{d_1} \ar[l]^-{d_0} & Y }$

1. A pair of morphisms $f,g \colon X \to Y$ in a model category are homotopic, written $f \sim g$ if they are both left and right homotopic.
2. A morphism $f \colon X \to Y$ in a model category is a homotopy equivalence is there is a morphism $h \colon Y \to X$ such that $hf \sim \mathrm{id}_ X$ and $fh \sim \mathrm{id}_ Y$.

The key feature of this discussion is that the notion of being homotopic is particularly nice in the case that the objects in question have good (co)fibrancy conditions.

Lemma 1.2.6 ([Corollary 1.2.6, Hov99]). Let $\mathcal{C}$ be a model category and $X,Y \in \mathcal{C}$. The property of being left (resp., right) homotopic is an equivalence relation on $\operatorname {Hom}(X,Y)$ for $X$ cofibrant and $Y$ fibrant and moreover the two notions coincide.

Proposition 1.2.7 ([Proposition 1.2.8, Hov99]). Let $\mathcal{C}$ be a model category, denote by $\mathcal{C}_{cf}$ the full subcategory of bifibrant objects. A morphism in $\mathcal{C}_{cf}$ is a weak equivalence if and only if it is a homotopy equivalence.

As we have seen that the homotopy relation is an equivalence relation on the morphisms when we restrict to the subcategory of bifibrant objects, and moreover it is compatible with composition, we can form the category $\mathcal{C}_{cf}/\sim$. This category is one of the fundamental objects in the theory of model categories.

Definition 1.2.8. Let $\mathcal{C}$ be a model category. Then its homotopy category $\operatorname {Ho}(\mathcal{C})$, up to equivalence of categories, is the category whose

• objects are the objects of $\mathcal{C}_{cf}$;
• morphisms are the homotopy classes of morphisms in $\mathcal{C}$. That is, the equivalence class of morphisms under homotopy.

Recall from the introduction that the overall goal is to construct the object $\mathcal{C}[W^{-1}]$ in such a way that we do not run into size issues. The following result tells us that the homotopy category as defined above is equivalent as a category to $\mathcal{C}[W^{-1}]$. It will therefore follow that $\mathcal{C}[W^{-1}]$ forms a category whenever $\mathcal{C}$ is a model category. Before we continue, let us give a formal definition of the object $\mathcal{C}[W^{-1}]$.

Definition 1.2.9. Let $\mathcal{C}$ be a category equipped with a class of weak equivalences. The object $\mathcal{C}[W^{-1}]$ is constructed in the following fashion. Begin by forming the free category $F(\mathcal{C},W^{-1})$ on the morphisms of $\mathcal{C}$ and the reversals of the morphisms of $W$. In particular an object of $F(\mathcal{C},W^{-1})$ is an object of $\mathcal{C}$ and a morphism is a finite string of composable morphisms $(f_1, f_2, \dots , f_ n)$ where $f_ i$ is either a morphism of $\mathcal{C}$ or the reversal $w_ i^{-1}$ of a morphism $w_ i \in W$. The empty string at a particular object plays the role of the identity at that object, and the composition is given by concatenation of strings.

One then defines $\mathcal{C}[W^{-1}]$ to be the quotient category of $F(\mathcal{C},W^{-1})$ by the following relations:

• $\operatorname {id}_ A = (\operatorname {id}_ A)$ for all objects $A$;
• $(f,g) = (g \circ f)$ for all composable morphisms $f,g$ of $\mathcal{C}$;
• $\operatorname {id}_{\operatorname {dom}w} = (w,w^{-1})$ for all $w \in W$;
• $\operatorname {id}_{\operatorname {codom}w} = (w^{-1},w)$ for all $w \in W$.

Note that a priori $\mathcal{C}[W^{-1}]$ need not be a category as one may need to use zig-zags of arbitrary length and as such the hom-objects need not be sets. The following theorem should be thought of as the fundamental theorem of model categories, it tells us that if $\mathcal{C}$ is a model category and $W$ is the class of weak equivalences, then $\mathcal{C}[W^{-1}]$ is in fact a category.

Theorem 1.2.10 ([Theorem 1.2.10, Hov99]). Let $\mathcal{C}$ be a model category, and $\gamma \colon \mathcal{C} \to \mathcal{C}[W^{-1}]$ the canonical functor.

• The inclusion $\mathcal{C}_{cf} \to \mathcal{C}$ induces an equivalence of categories $\operatorname {Ho}(\mathcal{C}) \simeq \mathcal{C}[W^{-1}]$ given by $\operatorname {Ho}(\mathcal{C}) \xrightarrow {\cong } \mathcal{C}_{cf}[W^{-1}] \to \mathcal{C}[W^{-1}]$.
• There are natural isomorphisms
\begin{align*} \operatorname {Hom}_\mathcal {C}(QRX, QRY)/\sim \, & \cong \operatorname {Hom}_{\mathcal{C}[W^{-1}]}(\gamma X , \gamma Y) \5pt]& \cong \operatorname {Hom}_\mathcal {C}(RQX, RQY)/\sim \, := [X,Y] \end{align*} where Q and R are the cofibrant and fibrant replacement functors respectively. In particular, \mathcal{C}[W^{-1}] is a category without moving to a higher universe and zig-zags of length at most three are all that are needed. • The functor \gamma \colon \mathcal{C} \to \mathcal{C}[W^{-1}] identifies left or right homotopic morphisms. • If f \colon A \to B is a morphism in \mathcal{C} such that \gamma f is an isomorphism in \mathcal{C}[W^{-1}] then f is a weak equivalence. Note that this theorem provides us with two equivalent ways of thinking about the homotopy category of a model category. On the one hand we can identify it with the subcategory of bifibrant objects with homotopy classes of morphismss, or we can alternatively see it as the category with the same objects as \mathcal{C} where we have formally inverted the weak equivalences. Remark 1.2.11. The above theorem tells us that when forming \mathcal{C}[W^{-1}] one needs zig-zags of length at most three. Explicitly, these zig-zags are of the form \[ X \leftarrow QX \xrightarrow {f} RY \leftarrow Y.

1.3 Examples

1.3.1 The Quillen model structure on $\textbf{Top}$

Definition 1.3.2. A map $f \colon X \to Y$ of topological spaces is said to be a weak homotopy equivalence if for each point $x \in X$ the map $f_\ast \colon \pi _ n (X,x) \to \pi _ n(Y,f(x))$ is a bijection of pointed sets for $n=0$, and an isomorphism of groups for $n \geq 1$.

Definition 1.3.3. A map $f \colon X \to Y$ of topological spaces is a Serre fibration if for each CW-complex $A$, the map $f$ has the RLP with respect to the inclusion $A \times 0 \to A \times [0,1]$.

Proposition 1.3.4 ([Theorem 2.4.19, Hov99]). There is a model structure on the category $\textbf{Top}$ where a map $f \colon X \to Y$ is a:

• weak equivalence if it is a weak homotopy equivalence;
• fibration if it is a Serre fibration;
• cofibration if it has the LLP with respect to all acyclic fibrations.

We will call this the Quillen model structure and denote it $\textbf{Top}_{\mathrm{Quillen}}$. All objects in this model structure are fibrant and the cofibrant objects are the retracts of relative cell complexes.

The homotopy category $\operatorname {Ho}(\textbf{Top}_\mathrm {Quillen})$ is the usual homotopy category, sometimes denoted $\textbf{hTop}$.

1.3.5 The projective model structure on $\textbf{Ch}_{\geq 0}(R)$

Proposition 1.3.6 ([Theorem 2.3.11, Hov99]). Let $R$ be a ring. Then there is a model structure on $\textbf{Ch}_{\geq 0}(R)$ where a morphism $f \colon X \to Y$ is a:

• weak equivalence if it is a quasi-isomorphism;
• fibration if it is a degreewise epimorphism in positive degree;
• cofibration if it is a degreewise monomorphism with projective cokernel.

We will call this the projective model structure and denote it $\textbf{Ch}_{\geq 0}(R)_\mathrm {proj}$. All objects are fibrant in this model structure. The cofibrant objects are those complexes which can be written as an increasing union of complexes such that the associated quotients are complexes of projectives with zero differential.

1.3.7 The natural model structure on $\textbf{Cat}$

• A functor $F \colon \mathcal{C} \to \mathcal{D}$ is an equivalence if it is fully faithful and essentially surjective.
• A functor $F \colon \mathcal{C} \to \mathcal{D}$ is an isofibration for all $c \in \mathcal{C}$ and any isomorphism $\phi \colon F(c) \cong d$, there exists an isomorphism $\psi \colon c \cong c'$ such that $F(\psi ) = \phi$.

Proposition 1.3.9 ([Theorem 3.1, Rez10]). There is a model structure on $\textbf{Cat}$ such that a functor $F \colon \mathcal{C} \to \mathcal{D}$ is a:

• weak equivalence if it is an equivalence;
• fibration if it is an isofibration;
• cofibration if the induced morphism $\operatorname {ob}F \colon \operatorname {ob}\mathcal{C} \to \operatorname {ob} \mathcal{D}$ is injective.

We will call this the natural model structure and denote it $\textbf{Cat}_\mathrm {Nat}$. All objects in this model structure are bifibrant.

1.3.10 Trivial model structures

Proposition 1.3.11 ([Example 1.1.5, Hov99]). Let $\mathcal{C}$ be a category with all small limits and colimits. Then there is a proper model structure on $\mathcal{C}$ where a morphism $f \colon X \to Y$ is a:

• weak equivalence if it is an isomorphism;
• fibration if it is any morphism;
• cofibration if it is any morphism.

We will call this the trivial model structure and denote it $\mathcal{C}_{\mathrm{triv}}$. All objects in this model structure are bifibrant.

In this case, from the universal property of the homotopy category, we have that $\operatorname {Ho}(\mathcal{C}) \cong \mathcal{C}$ as we are only inverting those morphisms which are already invertible.

In fact, the trivial model structure is one in a collection of three model structures which exist on a category $\mathcal{C}$ which possesses all small limits and colimits. One simply sets one of the three distinguished classes of morphisms to be the isomorphisms and then the other two are any morphisms of $\mathcal{C}$.

Proposition 1.3.12 ([Example 1.1.5, Hov99]). Let $\mathcal{C}$ be a category with all small limits and colimits. Then there is a model structure on $\mathcal{C}$ where a morphism $f \colon X \to Y$ is a:

• weak equivalence if it is any morphism;
• fibration if it is an isomorphism (resp., any morphism);
• cofibration if it is any morphism (resp., an isomorphism).

We will call this the terminal model structure (resp., initial structure) and denote it $\mathcal{C}_\mathrm {terminal}$ (resp., $\mathcal{C}_\mathrm {initial}$). In the fibration case only the terminal object is bifibrant, while in the cofibration case only the initial object is bifibrant. As such, in both cases, the homotopy category all objects become isomorphic.

1.5 References for Lecture 1

[BR13] Tobias Barthel and Emily Riehl. On the construction of functorial factorizations for model categories. Algebr. Geom. Topol., 13(2):1089-1124, 2013.

[DHKS04] William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith. Homotopy limit functors on model categories and homotopical categories, volume 113 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004.

[Hov99] Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI,1999.

[Joy] Andre Joyal. Joyal's CatLab.

[MSE] How was the functorial factorization axiom "frequently misstated"? Mathematics Stack Exchange.

[Qui69] Daniel G. Quillen. Rational homotopy theory. Ann. of Math. (2),90:205-295, 1969.

[Rez10] Charles Rezk. A model category for categories, 2010. Unpublished notes.

[Rie14] Emily Riehl. Categorical homotopy theory, volume 24 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2014.

[Rie19] Emily Riehl. Homotopical categories: From model categories to ($\infty$,1)-categories. arXiv:1904.00886v3, 2019.

[Wei01] Charles Weibel. Homotopy ends and Thomason model categories. Selecta Math. (N.S.), 7(4):533-564, 2001