Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
The Bousfield–Kan map(s) are comparison morphisms in a simplicial model category between two different “puffed up” versions of (co)limits over (co-)simplicial objects: one close to a homotopy (co)limit and the other a version of nerve/geometric realization.
Let be an (SSet,)-enriched category and write
for the canonical cosimplicial simplicial set (the adjunct of the hom-functor ).
Write furthermore for the fat simplex, the cosimplicial simplicial set which assigns to the nerve of the overcategory .
The Bousfield–Kan map of cosimplicial simplicial maps is a canonical morphism
of cosimplicial simplicial sets.
This can also be regarded as a morphism
This morphism induces the following morphisms between (co)simplicial objects in .
Bousfield–Kan for simplicial objects
For any simplicial object in , the realization of is the coend
where in the integrand we have the copower or tensor of by SSet.
Here the Bousfield–Kan map is the morphism
Bousfield–Kan for cosimplicial objects
For any cosimplicial object, its totalization is the -weighted limit
where in the integrand we have the power or cotensor of by SSet.
Here the Bousfield–Kan morphism is the morphism
Relation to homotopy limits
When the cosimplicial object is degreewise fibrant, then
computes the homotopy limit of as a weighted limit (as explained there). Then the above theorem says that the homotopy limit is already computed by the totalization
The original reference is
- Aldridge Bousfield and Dan Kan, Homotopy limits, completions and localizations Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.
- Hirschhorn, Simplicial model categories and their localization.
The Bousfield–Kan map(s) are on p. 397, def. 18.7.1 and def. 18.7.3.
Realization and totalization are defs 18.6.2 and 18.6.3 on p. 395.
Notice that this book writes for the nerve!