# nLab Bousfield-Kan map

model category

## Definitions

• category with weak equivalences

• weak factorization system

• homotopy

• small object argument

• resolution

• ## Universal constructions

• homotopy Kan extension

• Bousfield-Kan map

• ## Refinements

• monoidal model category

• enriched model category

• simplicial model category

• cofibrantly generated model category

• algebraic model category

• compactly generated model category

• proper model category

• stable model category

• ## Producing new model structures

• on functor categories (global)

• on overcategories

• Bousfield localization

• transferred model structure

• Grothendieck construction for model categories

• ## Presentation of $(\infty,1)$-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

• ## Model structures

• Cisinski model structure
• ### for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

• Strom model structure?
• Thomason model structure

• model structure on presheaves over a test category

• model structure on simplicial groupoids

• on cubical sets

• related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

• ### for $n$-groupoids

• for 1-groupoids

• ### for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

• ### for $\infty$-algebras

#### general

• on monoids

• on algebas over a monad

• on modules over an algebra over an operad

• #### specific

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

• model structure on dg-modules

• ### for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on presheaves of spectra

• ### for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

• ### for stable $(\infty,1)$-categories

• on dg-categories
• ### for $(\infty,1)$-operads

• on modules over an algebra over an operad

• ### for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

• ### for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

• model structure for (2,1)-sheaves/for stacks

• # Contents

## 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

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

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 $C$.

### Bousfield–Kan for simplicial objects

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

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

### Bousfield–Kan for cosimplicial objects

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

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

## Properties

###### Theorem

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.

###### 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!

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