Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
The fat simplex functor is a cosimplicial simplicial set
whose value at is a simplicial set that models the -simplex but is much bigger than the standard -simplex . This is such that is a cofibrant replacement of and of in the projective model structure on functors .
The fat simplex can be used to express the homotopy colimit over simplicial diagrams in terms of coends of the form . This construction is originally due to Bousfield and Kan.
Write for the simplex category. For write for the corresponding overcategory. Finally write
(in sSet) for the nerve of this overcategory.
This construction is functorial in :
There is a canonical morphism
of cosimplicial simplicial set, called the Bousfield-Kan map.
This exhibits as a cofibrant resolution of and of in the projective model structure on functors on .
See the discussion at Reedy model structure and at Bousfield-Kan map for details.
Revised on February 13, 2011 00:16:33
by Urs Schreiber