nLab Bousfield-Kan map

Idea

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.

Definition

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

$$\varphi \,\colon\, N\big(\Delta/(-)\big) \longrightarrow \Delta$$

of cosimplicial simplicial sets.

This can also be regarded as a morphism

$$\varphi \,\colon\, N\big( (-)/\Delta^{op} \big)^{op} \longrightarrow \Delta \,.$$

This morphism induces the following morphisms between (co)simplicial objects in $C$.

Bousfield–Kan for simplicial objects

For $X \,\colon\, \Delta^\op \to C$ any simplicial object in $C$, the realization of $X$ is the coend

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

where in the integrand we have the copower (or tensoring) of $C$ by sSet.

Here the Bousfield–Kan map is the morphism

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

Bousfield–Kan for cosimplicial objects

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

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

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

$$Tot X \simeq hom(\Delta,X) \stackrel {hom(\phi,Id_X)} {\longrightarrow} hom\Big( N\big(\Delta/(-)\ig) ,\, X \Big) \,.$$

Properties

Theorem

If the simplicial object $X$ is Reedy-cofibrant then its Bousfield–Kan map is a natural weak equivalence.

If the co-simplicial object $X$ is Reedy fibrant then its Bousfield–Kan map is a natural weak equivalence.

Proof

This can be proven for instance using homotopy colimits in the Reedy model structure. Details are at Reedy model structure – over the simplex category.

Relation to homotopy limits

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

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

References

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!