Lecture 2

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2.1 Things of Note

Remark 2.1.1. Recall that the framework of model categories is set up so that one can invert certain morphisms by taking the homotopy category. One can sometimes forget the fact that the property of being a weak equivalence is not a symmetric relation.

We define the pseudocircle $\mathbb {S}$ to be the (non-Hausdorff) topological space whose underlying set is the quadruple $\{ x_1, x_2, x_3, x_4\} $ and whose open subsets are

\[ \{ \{ x_1, x_2, x_3, x_4\} , \{ x_1, x_2, x_3\} , \{ x_1, x_2, x_4\} , \{ x_1, x_2\} , \{ x_1\} , \{ x_2\} , \emptyset \} . \]

There is a function of sets $f \colon S^1 \to \mathbb {S}$ which is defined by sending:

a point in the open left half of $S^1$ to the point $x_1$;
a point in the open right half of $S^1$ to the point $x_2$;
the top point of $S^1$ to $x_3$;
the bottom point of $S^1$ to $x_4$.

This function is a continuous function which is also a weak homotopy equivalence. However, the only continuous functions in the reverse direction are the constant maps which must induce the trivial map of $\pi _1$ [McC66].

Remark 2.1.2. Although a model category $\mathcal{C}$ has all small limits and colimits, it is rare for $\operatorname {Ho}(\mathcal{C})$ to be both complete and cocomplete. However, it is true that $\operatorname {Ho}(\mathcal{C})$ has all small products and coproducts as remarked in [Example 1.3.11, Hov99].

Here we will give an explicit example in $\operatorname {Ho}(\textbf{Top}_{\mathrm{Quillen}})$ of a pushout which does not exist.

We let $S^1 = \{ z \in \mathbb {C} \mid |z| = 1\} $ and consider the span

\[ \xymatrix{ S^1 \ar[r]^ f \ar[d]& S^1 \\ \ast & } \]

where $f(z) = z^2$. We assume that the pushout $P$ of the diagram exists. If such a $P$ exists, there would be a natural isomorphism $[P,X] = \operatorname {Hom}(\mathbb {Z}/2 , \pi _1(X))$.

Now, note that there is a fibration

\[ S^1 = B \mathbb {Z} \xrightarrow {f} S^1 \to \mathbb {R}P^\infty = B(\mathbb {Z}/2) \to \mathbb {C}P^\infty = BS^1 \xrightarrow {Bf} \mathbb {C}P^\infty \]

which gives an exact sequence

\[ [P,S^1] \to [P, \mathbb {R}P^\infty ] \to [P, \mathbb {C}P^\infty ]. \]

We of course know that $\pi _1(S^1) = \mathbb {Z}$, $\pi _1(\mathbb {R}P^\infty )= \mathbb {Z}/2$ and $\pi _1(\mathbb {C}P^\infty ) = 0$. Therefore if $[P,X] = \operatorname {Hom}(\mathbb {Z}/2 , \pi _1(X))$ then we have an exact sequence $0 \to \mathbb {Z}/2 \to 0$ which is impossible. As such, $P$ does not exist.

More examples and further discussion of this particular example can be found at [MOFa, MOFb].

2.2 Quillen Functors

Definition 2.2.1. Suppose that $\mathcal{C}$ and $\mathcal{D}$ are model categories. We say that a pair

\begin{equation*} F : \mathcal{C} \rightleftarrows \mathcal{D}: U \end{equation*}

of adjoint functors with $F$ the left adjoint is a Quillen adjunction if the following equivalent conditions are satisfied:

$F$ preserves cofibrations and acyclic cofibrations;
$U$ preserves fibrations and acyclic fibrations;
$F$ preserves cofibrations and $U$ preserves fibrations;
$F$ preserves acyclic cofibrations and $U$ preserves acyclic fibrations.

We say that $F$ is a left Quillen functor and $U$ is a right Quillen functor. The fact that these conditions are indeed equivalent can be found in [Proposition 8.5.3, Hir03].

Lemma 2.2.2 (Ken Brown's Lemma [Lemma 1.1.12, Hov99]). Given a Quillen adjunction $F : \mathcal{C} \rightleftarrows \mathcal{D}: U$ as above, then

$F$ preserves weak equivalences between cofibrant objects.
$U$ preserves weak equivalences between fibrant objects.

Lemma 2.2.3 ([Corollary A.2, Dug01]). Let $\mathcal{C}$ and $\mathcal{D}$ be model categories along with an adjoint pair $F : \mathcal{C} \rightleftarrows \mathcal{D}: U$. Then these functors are Quillen if and only if $U$ preserves fibrations between fibrant objects and acyclic fibrations.

Dually we have that these functors are Quillen if and only if $F$ preserves cofibrations between cofibrant objects and acyclic cofibrations.

Given a Quillen adjunction $F : \mathcal{C} \rightleftarrows \mathcal{D}: U$, we would like to get a derived adjunction (i.e., an adjunction of homotopy categories) $\mathbb {F} : \operatorname {Ho}(\mathcal{C}) \rightleftarrows \operatorname {Ho}(\mathcal{D}): \mathbb {U}$. Note that the following definition makes use of the choice of functorial factorization. Without functorial factorizations, the following construction would depend on a chosen cofibrant replacement (see [Section 8.5, Hir03]).

Definition 2.2.4. Let $F : \mathcal{C} \rightleftarrows \mathcal{D}: U$ be a Quillen adjunction between model categories, then we define

The left derived functor of $F$ to be the composite

\[ \mathbb {F} \colon \operatorname {Ho}(\mathcal{C}) \xrightarrow {\operatorname {Ho}(Q)} \operatorname {Ho}(\mathcal{C}) \xrightarrow {\operatorname {Ho}(F)} \operatorname {Ho}(\mathcal{D}). \]
The right derived functor of $U$ to be the composite

\[ \mathbb {U} \colon \operatorname {Ho}(\mathcal{D}) \xrightarrow {\operatorname {Ho}(R)} \operatorname {Ho}(\mathcal{D}) \xrightarrow {\operatorname {Ho}(U)} \operatorname {Ho}(\mathcal{C}). \]

The existence, and desired properties of these derived functors are the subject of [Section 1.3.2, Hov99]. Also, note by construction that one could define $\mathbb {F}$ even if $F$ is not a Quillen functor, it suffices that $F$ takes weak equivalences between cofibrant objects to weak equivalences.

Definition 2.2.5. Let $\mathcal{C}$ and $\mathcal{D}$ be model categories equipped with a Quillen adjunction $F : \mathcal{C} \rightleftarrows \mathcal{D}: U$. Then we say that $\mathcal{C}$ and $\mathcal{D}$ are Quillen equivalent if the derived adjunction $\mathbb {F} : \operatorname {Ho}(\mathcal{C}) \rightleftarrows \operatorname {Ho}(\mathcal{D}): \mathbb {U}$ is an equivalence of categories. We will sometimes write $\mathcal{C} \simeq _ Q \mathcal{D}$ to indicate the existence of a (potential zig-zag of) Quillen equivalences between two model structures.

In practice, one does not check Quillen equivalences in this fashion, instead there are some equivalent conditions that one usually checks instead.

Proposition 2.2.6 ([Proposition 1.3.13, Hov99]). Suppose we have a Quillen adjunction $F : \mathcal{C} \rightleftarrows \mathcal{D}: U$, then the following are equivalent:

The adjunction is a Quillen equivalence.
For all cofibrant $X \in \mathcal{C}$ and fibrant $Y \in \mathcal{D}$ a morphism $f \colon FX \to Y$ is a weak equivalence if and only if $\varphi (f) \colon X \to UY$ is a weak equivalence in $\mathcal{C}$.
The composite $X \xrightarrow {\eta } UFX \to URFX$ is a weak equivalence for all cofibrant $X$, and the composite $FQUX \to FUX \xrightarrow {\varepsilon } X$ is a weak equivalence for all fibrant $X$.

2.3 Stable Module Categories

Definition 2.3.1. A ring $R$ is Frobenius if the projective and injective $R$-modules coincide.

The main example of such a ring to keep in mind is the group ring $k[G]$ for $G$ a finite group and a field $k$. We now define the stable module category associated to a Frobenius ring.

Definition 2.3.2.

Let $R$ be a Frobenius ring, then two morphisms $f,g \colon M \to N$ in $ R\textbf{-Mod}$ are stably equivalent if the difference $f-g$ factors through a projective module.
Let $R$ be a Frobenius ring. Then the stable category of $R$-modules is the category whose objects are left $R$-modules and whose morphisms are the stable equivalence classes of $R$-module morphisms. We will denote the this category $\textbf{StMod}(R)$.

It should be suggestive that one can find a model structure on $ R\textbf{-Mod}$ such that the homotopy category of this model structure is equivalent to the stable module category, and indeed this is true.

Proposition 2.3.3 ([Theorem 2.2.12, Hov99]). Let $R$ be a Frobenius ring. Then there is a cofibrantly generated model structure on $R\textbf{-Mod}$ where a morphism $f \colon M \to N$ is a:

weak equivalence if it is a stable equivalence;
fibration if it is a surjection;
cofibration if it is an injection.

We will call this the stable model structure and denote it $ R\textbf{-Mod}_{\mathrm{st}}$. We will sometimes also refer to it as the model structure presenting the stable homotopy category as $\operatorname{Ho}(R\textbf{-Mod}_{\mathrm{st}}) \simeq \textbf{StMod}(R)$. All objects in this model structure are bifibrant.

Proposition 2.3.4 ([Sch02]). Let $p$ be an odd prime, then the homotopy categories of $(\mathbb {Z}/p)[\varepsilon ]/(\varepsilon ^2) \textbf{-Mod}_{\mathrm{st}}$ and $(\mathbb {Z}/p^2) \textbf{-Mod}_\mathrm {st}$ are triangulated equivalent, and are equivalent to $\textbf{StMod}(\mathbb {Z}/p)$. However there is no Quillen equivalence between these model structures.

2.4 Model Structures on $\textbf{Set}$

We will need notation for the following collections of morphisms in $\textbf{Set}$.

Definition 2.4.1.

$\operatorname {bij} :=$ the class of bijections.
$\operatorname {inj} :=$ the class of injections.
$\operatorname {surj} :=$ the class of surjections.
$\operatorname {all} :=$ the class of all morphisms.
$\operatorname {inj}_{\emptyset } :=$the class of injections with empty domain.
$\operatorname {inj}_{\neq \emptyset } :=$ the class of injections with non-empty domain.
$\operatorname {all}_{\neq \emptyset } :=$ the class of all morphisms with non-empty domain.

Proposition 2.4.2 ([BAC]). The following collections of morphisms give a model structure on the category $\textbf{Set}$ given as $(\operatorname {W},\operatorname {Fib},\operatorname {Cof})$:

$\textbf{Set}_{(-2)}^a$ := $(\operatorname {all},\operatorname {all},\operatorname {bij})$;
$\textbf{Set}_{(-2)}^b$ := $(\operatorname {all},\operatorname {inj},\operatorname {surj})$;
$\textbf{Set}_{(-2)}^c$ := $(\operatorname {all}, \operatorname {surj} \cup \operatorname {inj}_{\emptyset },\operatorname {inj}_{\neq \emptyset } \cup \{ \operatorname {id}_\emptyset \} )$;
$\textbf{Set}_{(-2)}^d$ := $(\operatorname {all}, \operatorname {bij} \cup \operatorname {inj}_{\emptyset },\operatorname {all}_{\neq \emptyset } \cup \{ \operatorname {id}_\emptyset \} )$;
$\textbf{Set}_{(-2)}^e$ := $(\operatorname {all}, \operatorname {surj}, \operatorname {inj})$;
$\textbf{Set}_{(-2)}^f$ := $(\operatorname {all}, \operatorname {bij}, \operatorname {all})$;
$\textbf{Set}_{(-1)}^a$ := $(\operatorname {all}_{\neq \emptyset } \cup \{ \operatorname {id}_\emptyset \} , \operatorname {surj} \cup \operatorname {inj}_{\emptyset }, \operatorname {inj})$;
$\textbf{Set}_{(-1)}^b$ := $(\operatorname {all}_{\neq \emptyset } \cup \{ \operatorname {id}_\emptyset \} , \operatorname {bij} \cup \operatorname {inj}_{\emptyset }, \operatorname {all})$;
$\textbf{Set}_{(0)}$ := $(\operatorname {bij},\operatorname {all},\operatorname {all})$.

Moreover this lift is exhaustive. That is, there are exactly nine model structures on the category of sets. The subscript of the model structure indicates a homotopy type (i.e., models with the same weak equivalences have the same subscript) while the superscript is an indexing.

Now that we have identified all of the possible model structures, all that is left is to describe the Quillen adjunctions between them. As the homotopy category is determined by the weak equivalences we see that there are three homotopy types. In particular, $(-2)$ refers to the fact that the homotopy category is equivalent to the terminal category, for $(-1)$ we have a discrete category on two objects, and the homotopy type of $(0)$ captures the homotopy type of sets (i.e., $\textbf{Set}_{(0)}$ is the trivial model structure).

To determine the Quillen equivalences, it is enough to determine if the identity functor is Quillen or not (indeed, every self-equivalence of $\textbf{Set}$ is naturally isomorphic to the identity functor).

There is a Quillen equivalence $\textbf{Set}_{(-1)}^{\, a} \rightleftarrows \textbf{Set}_{(-1)}^{\, b} $ where we follow our usual convention of the left Quillen functor being on top.

The situation for the models which model the homotopy theory of $(-2)$-types is slightly more complex. Indeed, there are no direct Quillen equivalences between all of the model structures. Instead, we have a zig-zag of a left and a right Quillen equivalence where there middle model is taken to be $\textbf{Set}_{(-2)}^{\, f}$ which has the maximal set of cofibrations. Equally, one could have factored through $\textbf{Set}_{(-2)}^{\, a}$ which has the maximal set of fibrations. The left Quillen functors between the $(-2)$-type models are given in the diagram below.

\[ \xymatrix@C=4em{ & b \ar[ddl] & \\ a \ar[ur] \ar[d] \ar[ddr] \ar[rr] \ar[drr] & & c \ar[d] \ar[dll] \ar[ddl]\\ f& & d \ar[ll]\\ & e \ar[ul]& } \]

2.5 Simplicial Sets

Definition 2.5.1. The simplex category $\Delta $ has for objects finite totally ordered sets $[n]=\{ 0,1, \dots , n\} $ and morphisms are generated by:

coface maps $\delta _ i \colon [n-1] \to [n]$ for $n > 0$ and $0 \leq i < n$, which is the injection whose image leaves out $i \in [n]$.
codegeneracy maps $\sigma _ i \colon [n+1] \to [n]$ for $n > 0$ and $0 \leq i < n$, which is the surjection such that $\sigma _ i(i) = \sigma _ i(i+1)=i$.

satisftying the simplicial identities.

Definition 2.5.2. A simplicial set is a functor $X \colon \Delta ^{op} \to \textbf{Set}$. We will denote by $\textbf{sSet}$ the category of all simplicial sets and natural transformations between them.

Using the definition of $\Delta $ from Definition 2.5.1, we are able to describe a simplicial set in a completely combinatorial fashion.

Definition 2.5.3. A simplicial set $X$ is a collection of sets $X_ n$ for all $n \geq 0$, and for each $n$ we have face maps $d_ i \colon X_ n \to X_{n-1}$ and degeneracy maps $s_ i \colon X_ n \to X_{n+1}$ satisfying the following relations:

\[ d_ i d_ j = d_{j-1} d_ i \qquad 0 \leq i < j \leq n \]

\[ s_ i s_ j= s_{j+1} s_{i} \qquad \; 0 \leq i \leq j < n \]

\[ d_ i s_ j = \begin{cases} s_{j-1} d_ i & \quad \qquad 0 \leq i < j < n \\ \text{id} & \qquad \quad 0 \leq j < n \text{ and } i=j \text{ or } i=j+1 \\ s_ j d_{i-1} & \qquad \quad 0 \leq j \text{ and } j+1 < i \leq n \end{cases} \]

Proposition 2.5.4 ([Example 1.5.5, Rie14]). Let $\mathcal{C}$ be a small category, then there is a simplicial set $N\mathcal{C}$, called the nerve of $\mathcal{C}$ defined such that:

$(N\mathcal{C})_0 = \operatorname {ob}(\mathcal{C})$;
$(N\mathcal{C})_1 = \operatorname {mor}(\mathcal{C})$;
$\cdots $;
$(N\mathcal{C})_ n = \{ \text{strings of }n \text{ composable arrows in }\mathcal{C}\} $.

The degeneracy map $s_ i$ inserts the identity at the $i^\text {th}$ position, and the face maps $d_ i$ compose the $i^\text {th}$ and $(i+1)^\text {st}$ arrows. The nerve is right adjoint of an adjoint pair

\[ \tau _1 : \textbf{sSet} \rightleftarrows \textbf{Cat} : N \]

The left adjoint $\tau _1$ is the functor that we call the fundamental category of a simplicial set.

Using the Yoneda embedding, we form the representable object $\Delta [n] = \operatorname {Hom}_\Delta (-,[n])$, which we call the standard $n$-simplex.

The key point about simplicial sets is that they provide a convenient combinatorial model for topological spaces. This correspondence is facilitated by a singular and geometric realization functor. In what follows, denote by $\Delta _ n \in \textbf{Top}$ be the standard topological $n$-simplex:

\[ \Delta _ n = \left\{ (x_0, \dots , x_ n) \in \mathbb {R}^{n+1} : 0 \leq x_ i \leq 1, \sum x_ i = 1\right\} . \]

Definition 2.5.5. Let $X$ be a topological space. The singular simplicial set of $X$ is the simplicial set $S(X)$ such that

\[ S(X)_ n = \text{Hom}_\textbf {Top}(\Delta _ n,X). \]

Dually to the above functor, we can define a functor $|-| \colon \textbf{sSet} \to \textbf{Top}$ called the geometric realization. For $\Delta [n]$ a representable object, we define $|\Delta [n]|$ to be $\Delta _ n$. One can then use a coend construction to extend this to the entire category.

Definition 2.5.6. Let $X$ be a simplicial set, then we define its geometric realization $|X|$ to be the topological space

\[ |X| := \int ^ n X_ n \times \Delta _ n. \]

Proposition 2.5.7. The pair of functors

\[ |-| : \textbf{sSet} \rightleftarrows \textbf{Top} : S(-) \]

form an adjoint pair.

Definition 2.5.8. The boundary of the standard $n$-simplex is the sub-simplicial set $\partial \Delta [n] \hookrightarrow \Delta [n]$ which is the union of all faces of $\Delta [n]$. We have that $|\partial \Delta [n]|$ is the boundary of the topological $n$-simplex [Remark 5.4, Rie11].

Definition 2.5.9. For each $n \geq 0$ and $0 \leq k \leq n$, the $(n,k)$-horn is the sub-simplicial set $\Lambda ^ k[n] \hookrightarrow \Delta [n]$ which is the union of all faces except the $k^\text {th}$ one. We say the horn is outer if $k=0,n$, and inner otherwise. We have that $|\Lambda ^ k[n]|$ is the union of all of the faces of the topological $n$-simplex with the $k^\text {th}$ face removed [Remark 5.4, Rie11].

Definition 2.5.10. A morphism of simplicial sets $f \colon X \to Y$ is a Kan fibration if it has the right lifting property for all horn inclusions. That is, we have the following commutative diagram for all $n \geq 0$ and $0 \leq k \leq n$:

\[ \xymatrix{\Lambda^k[n] \ar[d] \ar[r] & X \ar[d]^f \\ \Delta[n] \ar@{..>}[ur] \ar[r] & Y} \]

Definition 2.5.11. A morphism $f \colon X \to Y$ in $\textbf{sSet}$ is a monomorphism if it is levelwise injective. That is, for all $n \geq 0$ there is an injection of sets $f_ n \colon X_ n \to Y_ n$.

Proposition 2.5.12 ([Chapter II , Qui67]). There is a combinatorial model structure on $\textbf{sSet}$ where a morphism $f \colon X \to Y$ is a:

weak equivalence if its geometric realization $|f| \colon |X| \to |Y|$ is a weak homotopy equivalence in Top;
fibration is it is a Kan fibration;
cofibrations if it is a monomorphism.

We call this the Kan-Quillen model structure and denote it $\textbf{sSet}_\mathrm{Kan}$. The cofibrations are generated by the boundary inclusions $\partial \Delta [n] \to \Delta [n]$ for all $n \geq 0$ and the acyclic cofibrations are generated by the horn inclusions $\Lambda ^ k[n] \to \Delta [n]$ for all $n \geq 0$ and $0 \leq k \leq n$.

Definition 2.5.13. A fibrant object in $\textbf{sSet}_{\mathrm{Kan}}$ is called a Kan complex. In particular a Kan complex is a simplicial set $X$ such that we can find fillers

\[ \xymatrix{ \Lambda ^ k[n] \ar[d] \ar[r] & X \\ \Delta [n] \ar@{..>}[ur] } \]

for all $n \geq 0$ and $0 \leq k \leq n$.

Theorem 2.5.14 ([Chapter II , Qui67]). There is a Quillen equivalence

\[ |-| : \textbf{sSet}_\text {Kan} \rightleftarrows \textbf{Top}_{\mathrm{Quillen}} : S(-) \]

given by the geometric realization and singular functors where $\textbf{Top}_{\mathrm{Quillen}}$ is the Quillen model structure on $\textbf{Top}$.

Remark 2.5.15. Another useful feature of the Kan-Quillen model structure is that there is an explicit description of a functorial fibrant replacement functor, which we now recall [Kan57, Gui06].

Denote by $\operatorname {sd} \Delta [n]$ the barycentric subdivision of $\Delta [n]$ which is defined to be the nerve of the poset of subsets of $[n]$. For a simplicial set $X$, define the object $\text{Ex}(X) \in \textbf{sSet}$ as $\text{Ex}(X)_ n = \text{Hom}_\textbf {sSet}(\text{sd} \Delta [n] , X)$. This gives an endofunctor $\text{Ex} \colon \textbf{sSet} \to \textbf{sSet}$ which is right adjoint to the barycentric subdivision. There is a last vertex map $lv \colon \text{sd} \Delta [n] \to \Delta [n]$ induced via the poset morphism $(i_0, \dots , i_ m) \mapsto i_ m$. By taking colimits this gives a morphism $lv \colon \text{sd} X \to X$ for any simplicial set $X$. We denote by $j \colon X \to \text{Ex}(X)$ the adjoint to the last vertex map.

For an arbitrary simplicial set $X$, consider the directed system

\[ \xymatrix{X \ar[r]^-{j_ X} & \text{Ex}(X) \ar[r]^-{j_{\text{Ex}(X)}}& \text{Ex}^2(X) \ar[r]^-{j_{\text{Ex}^2(X)}}& \cdots } \]

whose colimit we denote $\text{Ex}^\infty (X)$. The functor $\text{Ex}^\infty (-)$ is a functorial fibrant replacement in the Kan-Quillen model structure on simplicial sets [Section III.5, Theorem 4.8, GJ99].

2.6 References for Lecture 2

[BAC] Tobias Barthel and Omar Antolin-Camarena The nine model category structures on the category of sets.

[Dug01] Daniel Dugger. Replacing model categories with simplicial ones. Trans. Amer. Math. Soc., 353(12):5003-5027), 2001.

[GJ99] Paul Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhauser Verlag, Basel,1999.

[Gui06] B. Guillou. Kan's $\operatorname{Ex}^\infty$ functor, 2007. Unpublished notes.

[Hir03] Philip S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI,2003.

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

[Kan57] Daniel M. Kan. On c. s. s. complexes. Amer. J. Math., 79:449-476, 1957.

[McC66] Michael C. McCord. Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J., 33:465-474, 1966.

[MOFa] Categorical homotopy colimits. MathOverflow.

[MOFb] The homotopy category is not complete nor cocomplete. MathOverflow.

[Qui67] Daniel G. Quillen. Homotopical Algebra, volume 43 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1967.

[Rie11] Emily Riehl. A leisurely introduction to simplicial sets, 2011.

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

[Sch02] Marco Schlichting. A note on $K$-theory and triangulated categories. Invent. Math., 150(1):111-116, 2002.