nLab
geometric realization of categories

Definition

Write

$$N:\mathrm{Cat}\to \mathrm{sSet}$$N : Cat \to sSet

for the nerve functor from Cat to sSet. Write

$$\mid -\mid :\mathrm{sSet}\to \mathrm{Top}$${\vert - \vert} : sSet \to Top

for the geometric realization of simplicial sets.

The geometric realization of categories is the composite

$$\mid -\mid :=\mid N(-)\mid :\mathrm{Cat}\to \mathrm{Top}$${\vert - \vert} := {\vert N(-)\vert} : Cat \to Top

Properties

Thomason model structure

There is a model category structure on Cat whose weak equivalences are those functors which under geometric realization are weak equivalences in the standard model structure on topological spaces: the Thomason model structure.

Recognizing weak equivalences: Quillens theorem A and B

Let $F:C\to D$ be a functor.

Theorem

(Quillen’s theorem A)

If for all objects $d\in D$ the geometric realization $\mid F/d\mid$ of the comma category $F/d$ is contractible (meaning that $F$ is a “homotopy cofinal functor”, hence a cofinal (∞,1)-functor), then $\mid f\mid :\mid C\mid \to \mid D\mid$ is a weak homotopy equivalence.

Theorem

(Quillen’s theorem B)

If for all $d\in D$ we have that $\mid F/d\mid$ is weakly homotopy equivalent to a given topological space $X$ and all morphisms $f:d_1\to d_2$ induce weak homotopy equivalences between these, then $X$ is the homotopy fiber of $\mid F\mid$, hence we have a fiber sequence

$$X\to \mid C\mid \stackrel{\mid F\mid }{\to }\mid D\mid .$$X \to \vert C \vert \stackrel{\vert F \vert }{\to} \vert D \vert \,.

Natural transformations and homotopies

Proposition

A natural transformation $\eta :F\Rightarrow G$ between two functors $F,G:C\to D$ induces under geometry realization a homotopy $\mid \eta \mid :\mid F\mid \to \mid G\mid$.

Proof

The natural transformation is equivalently a functor

$$\eta :C\times \{0\to 1\}\to D.$$\eta : C \times \{0 \to 1\} \to D \,.

SInce geometric realization of simplicial sets preserves products (see there) we have that $\mid C\times \{0,1\}\mid {\simeq }_{\mathrm{iso}}\mid C\mid \times \mid \{0\to 1\}\mid$. But this is a cylinder object in topological spaces, hence $\mid \eta \mid$ is a left homotopy.

Corollary

An equivalence of categories $C\simeq D$ induces a homotopy equivalence between their geometric realizations.

Remark

Notice that the converse is far from true: very different categories can have geometric realizations that are (weakly) homotopy equivalent. This is because geometric realization implicitly involves Kan fibrant replacement: it freely turns morphisms into equivalences.

Corollary

If a category $C$ has an initial object or a terminal object, then its geometric realization is contractible.

Proof

Assume the case of a terminal object, the other case works dually. Write $*$ for the terminal category.

Then we have an equality of functors

$${\mathrm{Id}}_{*}=(*\stackrel{\perp }{\to }C\to *),$$Id_* = (* \stackrel{\bottom}{\to} C \to *) \,,

where the first functor on the right picks the terminal object, and we have a natural transformation

$${\mathrm{Id}}_C\Rightarrow (C\to *\stackrel{\perp }{\to }C)$$Id_C \Rightarrow (C \to * \stackrel{\bottom}{\to} C)

whose components are the unique morphisms into the terminal object.

By prop. 1 it follows that we have a homotopy equivalence $\mid C\mid \to \mid *\mid =*$.

Categories and posets

Definition

For $C$ a category, let $\nabla C$ be the poset of simplices in $NC$, ordered by inclusion. Its nerve is also called the barycentric subdivision of the nerve of $C$.

Proposition

For every category $C$ the poset $\nabla C$ has equivalent geometric realization

$$\mid \nabla C\mid \simeq \mid C\mid .$$\vert \nabla C \vert \simeq \vert C \vert \,.

Behaviour under homotopy colimits

Proposition

For $F:D\to$ Cat a functor, let $\mid F(-)\mid :D\stackrel{F}{\to }\mathrm{Cat}\stackrel{\mid -\mid }{\to }$ Top be the postcomposition with geometric realization.

Then we have a weak homotopy equivalence

$$\mid \int F\mid \simeq \mathrm{hocolim}\mid F(-)\mid$$\vert \int F \vert \simeq hocolim \vert F(-) \vert

exhibiting the homotopy colimit in Top over $\mid F(-)\mid$ as the geometric realization of the Grothendieck construction $\int F$ of $F$.

This is due to (Thomason).

Related concepts

• geometric realization

  • of categories, of simplicial topological spaces, of cohesive ∞-groupoids

References

For general references see also nerve and geometric realization.

Quillen’s theorems A and B and their generalizations are discussed for instance in

• Jonathan Ariel Barmak, On Quillen’s Theorem A for posets (arXiv:1005.0538)

• Clark Barwick, Dan Kan, A Quillen theorem $B_n$ for homotopy pullbacks (arXiv:1101.4879)

• David Roberts, Theorem A for topological categories

The geometric realization of Grothendieck constructions has been analyzed in

• R. W. Thomason, Homotopy colimits in the category of small categories , Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91109.