Section 3. (0MT0) · ScalingStacks

3. Nerve constructions and classification diagrams[0MT0]

In this section we discuss a construction called the classification diagram, which produces a simplicial space from a pair of categories. A special case of this construction of particular interest is the classifying diagram of a category, which produces a full embedding $N\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ of the category of small categories into the category of simplicial spaces, which has the property that $N$ takes equivalences of categories, and only equivalences of categories, to weak equivalences of simplicial spaces. Another special case of this construction is the application of the classification diagram to model categories, which will be considered in Section 8.

In what follows we write $D^{C}$ for the category of functors from $C$ to $D$ .

3.1. The nerve of a category[0MT1]

Given a category $C$ , let $\nerve C$ denote the nerve of $C$ ; that is, $\nerve C$ is a simplicial set whose $n$ -simplices consist of the set of functors $[n]\rightarrow C$ . (The classifying space $BC$ of a category is a topological space which is the geometric realization of the nerve.) The following is well-known.

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Proposition 3.2. The nerve of $[n]$ is $\Delta[n]$ . For categories $C$ and $D$ there are natural isomorphisms

\nerve(C\times D)\approx\nerve C\times\nerve D\qquad\text{and}\qquad\nerve(D^{C})\approx\nerve(D)^{\nerve(C)}.

The functor $\nerve\colon{\operatorname{\mathcal{C}at}}\rightarrow{\operatorname{\mathcal{S}}}$ is a full embedding of categories. Furthermore, if $C$ is a groupoid then $\nerve(C)$ is a Kan complex.

Although the nerve functor is a full embedding, it is awkward from our point of view, since non-equivalent categories may give rise to weakly equivalent nerves.

3.3. The classification diagram of a pair of categories[0MT2]

Consider a pair $(C,W)$ consisting of a category $C$ together with a subcategory $W$ such that ${\operatorname{ob}}W={\operatorname{ob}}C$ ; we refer to a morphism of $C$ as a weak equivalence if it is contained in $W$ . More generally, given a natural transformation $\alpha\colon f\rightarrow g$ of functors $f,g\colon D\rightarrow C$ , we say that $\alpha$ is a weak equivalence if $\alpha d\in W$ for each $d\in{\operatorname{ob}}D$ , and write $\we(C^{D})$ for the category consisting of all functors from $D$ to $C$ and all weak equivalences between them; thus $\we(C)=W$ .

For any such pair $(C,W)$ of categories we define a simplicial space $N(C,W)$ , called the classification diagram of $(C,W)$ , by setting

N(C,W)_{m}=\nerve\we(C^{[m]}).

If we view the category $[m]\times[n]$ as an $m$ -by- $n$ grid of objects with rows of $m$ composable horizontal arrows and columns of $n$ composable vertical arrows, then the set of $n$ -simplices of the $m$ th space of $N(C,W)$ corresponds to the set of functors $[m]\times[n]\rightarrow C$ in which the vertical arrows are sent into $W\subset C$ .

We consider several special cases of this construction.

3.4. Discrete nerve construction[0MT3]

A special case of the classification diagram is the discrete nerve. Let $C_{0}\subset C$ denote the subcategory of $C$ consisting of all its objects and only identity maps between them, and let $\discnerve C=N(C,C_{0})$ . Note that $\nerve C=\diag(\discnerve C)$ , and that $\discnerve([n])=F(n)$ .

It is not hard to see that the functor $\discnerve\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ embeds the category of small categories as a full subcategory of simplicial spaces. The discrete nerve functor is awkward from our point of view, since equivalent categories can have non-weakly equivalent discrete nerves.

3.5. The classifying diagram of a category[0MT4]

We give a construction which embeds the category of categories inside the category of simplicial spaces and which carries equivalences of categories (and only equivalences) to weak equivalences of simplicial spaces.

Given a category $C$ , define a simplicial space $NC=N(C,\iso C)$ , where $\iso C\subset C$ denotes the maximal subgroupoid. Thus, the $m$ th space of $NC$ is $(NC)_{m}=\nerve\iso(C^{[m]})$ . We call $NC$ the classifying diagram of $C$ .

Let $I[n]$ denote the category having $n+1$ distinct objects, and such that there exists a unique isomorphism between any two objects. We suppose further that there is a chosen inclusion $[n]\rightarrow I[n]$ . Then the set of $n$ -simplices of the $m$ th space of $NC$ corresponds to the set of functors $[m]\times I[n]\rightarrow C$ . Note that there is a natural isomorphism

(NC)_{m}\approx(NC)_{1}\times_{(NC)_{0}}\dots\times_{(NC)_{0}}(NC)_{1}

(3.6)

(where the right-hand side is an $m$ -fold fiber-product), and that the natural map $(d_{1},d_{0})\colon(NC)_{1}\rightarrow(NC)_{0}\times(NC)_{0}$ is a simplicial covering space, with fiber over any vertex $(x,y)\in(NC)_{0}^{2}$ naturally isomorphic to the set $\hom_{C}(x,y)$ .

If the category $C$ is a groupoid, then the natural map $\nerve C\rightarrow NC$ , where $\nerve C$ is viewed as a constant simplicial space, is a weak equivalence; this follows from the fact that for $C$ a groupoid, $\iso(C^{[m]})$ , $C^{[m]}$ , and $C$ , are equivalent categories. It is therefore natural to regard the classifying diagram construction as a generalization of the notion of a classifying space of a groupoid.

The following theorem says that $N\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ is a full embedding of categories which preserves internal hom-objects, and furthermore takes a functor to a weak equivalence if and only if it is an equivalence of categories.

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Theorem 3.7. Let $C$ and $D$ be categories. There are natural isomorphisms

N(C\times D)\approx NC\times ND\qquad\text{and}\qquad N(D^{C})\approx(ND)^{NC}

of simplicial spaces. The functor $N\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ is a full embedding of categories. Furthermore, a functor $f\colon C\rightarrow D$ is an equivalence of categories if and only if $Nf$ is a weak equivalence of simplicial spaces.

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Proof. That $N$ preserves products is clear.

To show that $N(D^{C})\rightarrow(ND)^{NC}$ is an isomorphism, we must show that for each $m,n\geq 0$ this map induces a one-to-one correspondence between functors $[m]\times I[n]\rightarrow D^{C}$ and maps $F(m)\times\Delta[n]\rightarrow(ND)^{NC}$ . By (3.8) it will suffice to show this for the case $m=n=0$ ; that is, to show that functors $C\rightarrow D$ are in one-to-one correspondence with maps $NC\rightarrow ND$ , or in other words, that $N\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ is a full embedding of categories.

To see that $N$ is a full embedding, note that any map $NC\rightarrow ND$ is determined by how it acts on the $0$ th and $1$ st spaces of $NC$ . The result follows from a straightforward argument using (3.2) and the fact that $(d_{1},d_{0})$ is a simplicial covering map such that $d_{1}s_{0}=1=d_{0}s_{0}$ for both $NC$ and $ND$ .

It is immediate that naturally isomorphic functors induce simplicially homotopic maps of simplicial spaces since $N(C^{I[1]})\approx(NC)^{\Delta[1]}$ by (3.8), and thus an equivalence of categories induces a weak equivalence of simplicial spaces. To prove the converse, note that (3.9) will show that $(ND)^{NC}\approx N(D^{C})$ is Reedy fibrant, and in particular $\Map_{{\operatorname{\mathcal{S}}}}(NC,ND)\approx(ND^{NC})_{0}$ is a Kan complex. Therefore, if $Nf\colon NC\rightarrow ND$ is a weak equivalence of simplicial spaces it must be a simplicial homotopy equivalence. Furthermore, the homotopy inverse is a $0$ -simplex of $N(C^{D})_{0}$ and the simplicial homotopies are $1$ -simplices of $N(D^{C})_{0}$ and $N(C^{D})_{0}$ ; by what we have already shown these correspond precisely to a functor $g\colon D\rightarrow C$ and natural isomorphisms $fg\sim 1_{D}$ and $gf\sim 1_{C}$ , as desired. ∎

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Lemma 3.8. Let $C$ be a category. Then there are natural isomorphisms

N([m]\times C)\approx F(m)\times NC\qquad\text{and}\qquad N(C^{I[n]})\approx(NC)^{\Delta[n]}

of simplicial spaces.

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Proof. The first isomorphism follows from the fact that $N$ preserves products and that $N([m])\approx F(m)$ . The second isomorphism may be derived from the fact that $\iso(D^{I[n]})\approx(\iso D)^{I[n]}\approx(\iso D)^{[n]}$ for any category $D$ , and thus in particular when $D=C^{[m]}$ . ∎

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Lemma 3.9. If $C$ is a category, then $NC$ is a Reedy fibrant simplicial space.

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Proof. We must show that

\ell_{n}\colon(NC)_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(F(n),NC)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(\dot{F}(n),NC)

is a fibration for each $n\geq 0$ . We have the following cases:

$n=0$ :: $(NC)_{0}=\nerve(\iso C)$ is a Kan complex by (3.2).
$n=1$ :: $\ell_{1}\colon(NC)_{1}\rightarrow(NC)_{0}\times(NC)_{0}$ is a simplicial covering space with discrete fiber, and thus is a fibration.
$n=2$ :: $\ell_{2}$ is isomorphic to an inclusion of path-components, and so is a fibration.
$n\geq 3$ :: $\ell_{n}$ is an isomorphism, and thus a fibration.

∎

3.10. Classification diagrams of functor categories[0MT5]

The following generalizes one of the statements of (3.7), and we note it for future reference.

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Proposition 3.11. Let $C$ and $D$ be categories, and $W\subset D$ a subcategory such that $\iso D\subset W$ . Then there are natural isomorphisms

N(D^{C},\we(D^{C}))\approx N(D,W)^{NC}\approx N(D,W)^{\discnerve C}.

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Proof. We must show that for each $m,n\geq 0$ the natural maps $N(D^{C},\we(D^{C}))\rightarrow N(D,W)^{NC}\rightarrow N(D,W)^{\discnerve C}$ induce one-to-one correspondences amongst the sets of

(1)

functors $[m]\times[n]\rightarrow D^{C}$ which carry “vertical” maps into $\we(D^{C})$ ,
(2)

maps $F(m)\times\Delta[n]\rightarrow N(D,W)^{NC}$ of simplicial spaces, and
(3)

maps $F(m)\times\Delta[n]\rightarrow N(D,W)^{\discnerve C}$ of simplicial spaces.

By (3.8) and (3.12) it will suffice to show this in the case $m=n=0$ , in which case the result becomes a straightforward computation. ∎

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Lemma 3.12. Let $C$ a category, and $W$ a subcategory with ${\operatorname{ob}}W={\operatorname{ob}}C$ . Then there is a natural isomorphism

N(\widetilde{C^{[n]}},\we(\widetilde{C^{[n]}}))\approx(NC)^{\Delta[n]},

where $\widetilde{C^{[n]}}\subset C^{[n]}$ denotes the full subcategory whose objects are those functors $[n]\rightarrow C$ which factor through $W\subset C$ , and $\we(\widetilde{C^{[n]}})=\we(C^{[n]})\cap\widetilde{C^{[n]}}$ .

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Proof. For any pair $(D,W)$ of category $D$ and subcategory $W$ , we have that $\we(\widetilde{D^{[n]}})=W^{[n]}$ , and that $\nerve(W^{[n]})=(\nerve W)^{\Delta[n]}$ . We obtain the result by substituting $C^{[m]}$ for $D$ . ∎

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3