Paths and cylinders
for the nerve functor from Cat to sSet. Write
for the geometric realization of simplicial sets.
The geometric realization of categories is the composite
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 be a functor.
Natural transformations and homotopies
A natural transformation between two functors induces under geometry realization a homotopy .
The natural transformation is equivalently a functor
SInce geometric realization of simplicial sets preserves products (see there) we have that . But this is a cylinder object in topological spaces, hence is a left homotopy.
Assume the case of a terminal object, the other case works dually. Write for the terminal category.
Then we have an equality of functors
where the first functor on the right picks the terminal object, and we have a natural transformation
whose components are the unique morphisms into the terminal object.
By prop. 1 it follows that we have a homotopy equivalence .
Categories and posets
For a category, let be the poset of simplices in , ordered by inclusion.
For every category the poset has equivalent geometric realization
Behaviour under homotopy colimits
For Cat a functor, let Top be the postcomposition with geometric realization.
Then we have a weak homotopy equivalence
exhibiting the homotopy colimit in Top over as the geometric realization of the Grothendieck construction of .
This is due to (Thomason).
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.