Bousfield-Kan map

cartesian closed model category, locally cartesian closed model category

- Strom model structure?

on chain complexes/model structure on cosimplicial abelian groups

related by the Dold-Kan correspondence

model structure on differential graded-commutative superalgebras

on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations

model structure for (2,1)-sheaves/for 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 $C$ be an (SSet,$\otimes = \times$)-enriched category and write

$\Delta : \Delta \to Set^{\Delta^{op}}$

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

$\varphi : N(\Delta/(-)) \to \Delta$

of cosimplicial simplicial sets.

This can also be regarded as a morphism

$\varphi : N((-)/\Delta^{op})^{op} \to \Delta
\,.$

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

$|X| := X \otimes_{\Delta^{op}} \Delta :=
\int^{[n] \in \Delta} X_n \otimes \Delta^n
\,,$

where in the integrand we have the copower or tensor of $C$ by SSet.

Here the **Bousfield–Kan** map is the morphism

$X \otimes_{\Delta^{op}} N((-)/\Delta^{op})^{op}
\stackrel{Id_X \otimes_{\Delta^{op}} \phi }{\to}
X \otimes_{\Delta^{op}} \Delta
\,.$

For $X : \Delta \to C$ any cosimplicial object, its **totalization** is the $\Delta$-weighted limit

$Tot X := lim^\Delta X \simeq \int_{[n \in \Delta]} X_n^{\Delta^n}
\,,$

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

$Tot X \simeq hom(\Delta,X) \stackrel{hom(\phi,Id_X)}{\to}
hom(N(\Delta/(-)), X)
\,.$

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

$lim^{N(\Delta/(-))} X
\simeq holim X$

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

$holim X \simeq lim^\Delta X
\,.$

The original reference is

- Aldridge Bousfield and Dan Kan,
*Homotopy limits, completions and localizations*Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.

Reviews include

- 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 $B$ for the nerve!

Last revised on September 19, 2011 at 13:20:06. See the history of this page for a list of all contributions to it.