category theory

Contents

Definition

Write

$N : Cat \to sSet$

for the nerve functor from Cat to sSet. Write

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

for the geometric realization of simplicial sets.

The geometric realization of categories is the composite

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

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 $\vert F/d\vert$ of the comma category $F/d$ is contractible (meaning that $F$ is a “homotopy cofinal functor”, hence a cofinal (∞,1)-functor), then $\vert F \vert : \vert C \vert \to \vert D \vert$ is a weak homotopy equivalence.

Theorem

(Quillen’s theorem B)

If for all $d \in D$ we have that $\vert F/d\vert$ 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 $\vert F \vert$, hence we have a fiber sequence

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

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 $|\eta| : \vert F\vert \to \vert G \vert$.

Proof

The natural transformation is equivalently a functor

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

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

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

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

$Id_C \Rightarrow (C \to * \stackrel{\bottom}{\to} C)$

whose components are the unique morphisms into the terminal object.

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

Categories and posets

Definition

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

Proposition

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

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

Behaviour under homotopy colimits

Proposition

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

Then we have a weak homotopy equivalence

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

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

This is due to (Thomason).