model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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 $C$ be an (SSet,$\otimes = \times$)-enriched category and write
for the canonical cosimplicial simplicial set (the adjunct of the hom-functor $\Delta^{op} \times \Delta \to Set$).
Write furthermore $N(\Delta/-) : \Delta \to Set^{\Delta^{op}}$ for the fat simplex, the cosimplicial simplicial set which assigns to $[n]$ the nerve of the overcategory $\Delta / [n]$.
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 $C$.
For $X : \Delta^\op \to C$ any simplicial object in $C$, the realization of $X$ is the coend
where in the integrand we have the copower or tensor of $C$ by SSet.
Here the Bousfield–Kan map is the morphism
For $X : \Delta \to C$ any cosimplicial object, its totalization is the $\Delta$-weighted limit
where in the integrand we have the power or cotensor $X_n^{\Delta^n} = \pitchfork(\Delta, X_n)$ of $C$ by SSet.
Here the Bousfield–Kan morphism is the morphism
If the simplicial object $X$ is Reedy cofibrant then its Bousfield–Kan map is a natural weak equivalence.
If the cosimplicial object $X$ is Reedy fibrant then its Bousfield–Kan map is a natural weak equivalence.
This can be proven for instance using homotopy colimits in the Reedy model structure. Details are at Reedy model structure – over the simplex category .
When the cosimplicial object $X$ is degreewise fibrant, then
computes the homotopy limit of $X$ 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
Reviews include
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 $B$ for the nerve!