nLab
geometric realization of categories

Context

Homotopy theory

Category theory

Contents

Definition

Write

N:CatsSet N : Cat \to sSet

for the nerve functor from Cat to sSet. Write

||:sSetTop {\vert - \vert} : sSet \to Top

for the geometric realization of simplicial sets.

The geometric realization of categories is the composite

||:=|N()|:CatTop {\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:CDF : C \to D be a functor.

Theorem

(Quillen’s theorem A)

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

Theorem

(Quillen’s theorem B)

If for all dDd \in D we have that |F/d|\vert F/d\vert is weakly homotopy equivalent to a given topological space XX and all morphisms f:d 1d 2f : d_1 \to d_2 induce weak homotopy equivalences between these, then XX is the homotopy fiber of |F|\vert F \vert, hence we have a fiber sequence

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

Natural transformations and homotopies

Proposition

A natural transformation η:FG\eta : F \Rightarrow G between two functors F,G:CDF, G : C \to D induces under geometry realization a homotopy |η|:|F||G||\eta| : \vert F\vert \to \vert G \vert.

Proof

The natural transformation is equivalently a functor

η:C×{01}D. \eta : C \times \{0 \to 1\} \to D \,.

SInce geometric realization of simplicial sets preserves products (see there) we have that |C×{0,1}| iso|C|×|{01}|\vert C \times \{0,1\}\vert \simeq_{iso} \vert C \vert \times \vert \{0 \to 1\} \vert. But this is a cylinder object in topological spaces, hence |η|\vert \eta \vert is a left homotopy.

Corollary

An equivalence of categories CDC \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 CC 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

Id *=(*C*), Id_* = (* \stackrel{\bottom}{\to} C \to *) \,,

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

Id C(C*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 |C||*|=*\vert C \vert \to \vert * \vert = *.

Categories and posets

Definition

For CC a category, let C\nabla C be the poset of simplices in NCN C, ordered by inclusion.

Proposition

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

|C||C|. \vert \nabla C \vert \simeq \vert C \vert \,.

Behaviour under homotopy colimits

Proposition

For F:DF : D \to Cat a functor, let |F()|:DFCat||\vert F(-)\vert : D \stackrel{F}{\to} Cat \stackrel{\vert-\vert}{\to} Top be the postcomposition with geometric realization.

Then we have a weak homotopy equivalence

|F|hocolim|F()| \vert \int F \vert \simeq hocolim \vert F(-) \vert

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

This is due to (Thomason).

References

For general references see also nerve and geometric realization.

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

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.

Revised on December 11, 2013 11:39:41 by Anonymous Coward (82.82.242.221)