nLab Bousfield-Kan formula

Idea

The Bousfield-Kan formula (BK72) is a presentation of colimits (or dually, limits) in $(\infty,1)$-categories (i.e., homotopy colimits) in terms of coproducts and colimits of simplicial objects.

In ordinary category theory, the colimit of a diagram $F \colon I\to C$ may be computed as a coequalizer of the parallel pair

$$\coprod_{c_{0}\to c_{1}}F c_{0} \rightrightarrows \coprod_{c}F c \,.$$

When $C$ is an $(\infty,1)$-category, we need to take “higher data” into account. So we take the geometric realization (i.e., $\Delta^{op}$-indexed colimit) of the simplicial object

$$B_{\bullet}(\ast,I,F) \;=\; \left( \cdots \coprod_{c_{0}\to c_{1}\to c_{2}} F c_{0} \Rrightarrow \coprod_{c_{0}\to c_{1}} F c_{0} \rightrightarrows \coprod_{c\in C} F c \right) \,.$$

This can be formalized as follows: There is an “initial vertex” functor $\mathrm{init} \colon (\Delta_{/I})^{op}\to I$ from the opposite of the category of simplices of $I$, which is homotopy final. Thus,

$$\mathrm{colim}_{I} F \;\simeq\; \mathrm{colim}_{(\Delta_{/I})^{op}}(F\circ\mathrm{init}) \,.$$

The simplicial object $B_{\bullet}(\ast,I,F)$ is just the left Kan extension of $F\circ\mathrm{init}$ along the projection $(\Delta_{/I})^{op}\to\Delta^{op}$, so we have

$$\mathrm{colim}_{(\Delta_{/I})^{op}} (F\circ\mathrm{init}) \simeq \mathrm{colim}_{\Delta^{op}}B_{\bullet}(\ast,I,F) \,.$$

Particular Implementations in Model Categories

Via Geometric Realizations

In a simplicial model category $C$, homotopy colimits indexed by $\Delta^{op}$ are modeled by geometric realization (see there). Therefore, the Bousfield-Kan formula for the homotopy colimit of a (pointwise cofibrant) diagram $F \colom I\to C$ is computed by

$${\mathrm {hocolim}}_{I}F \simeq {\big|B_{\bullet}(\ast,I,F)\big|} \,.$$

Textbook account includes Rie14, Chapter 5.

Via Functor Tensor Product

Let $F \colon I\to C$ be a (pointwise cofibrant) diagram in a simplicial model category. Some coend calculus shows Rie14, 6.6.1 that

$$\big\vert B_{\bullet}(\ast,I,F) \big\vert \;\cong\; \int^{i\in I}N(I_{/i})\otimes F(i) \;\eqqcolon\; N(I_{/-})\otimes_{I}F \,.$$

The right hand side is also called the Bousfield-Kan formula.

In Model Categories with Framings

A cosimplicial framing on a model category is roughly a functorial choice of cosimplicial resolutions Hir03, Chapter 16. With this data, we can formally mimic the Bousfield-Kan formula, and this is adopted as the definition of homotopy colimits in Hir03, Chapter19; that this approach is equivalent to the ordinary notion of homotopy colimits is established in AO23 (for combinatorial model categories) and Ara23, Theorem 1.16 (for general model categories).

Bousfield–Kan map

In a simplicial model category (this can be generalized, as discussed above), there are two formulas for homotopy colimits of simplicial objects: The Bousfield–Kan formula, and geometric realization. The Bousfield–Kan map(s) compare these formulas.

More precisely, 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

There is also the Bousfield–Kan map 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

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

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.

Example

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 Bousfield-Kan formula was introduced in:

• Aldridge K. Bousfield and Daniel M. Kan, Chapter IX in: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304 (1972; 1987), Springer (doi:10.1007/978-3-540-38117-4)

Discussion in model categories:

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf, pdf]

  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.

  Beware that this book writes $B$ for the nerve!

Discussions in simplicial model categories:

• Emily Riehl, Chapter 5 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

See also:

• Sergey Arkhipov, Sebastian Ørsted: Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories, Appl. Categ. Structures 31 6 (2023) [doi:10.1007/s10485-023-09747-8]

• Kensuke Arakawa, Section 1 of: Homotopy Limits and Homotopy Colimits of Chain Complexes, (2023) [arXiv:2310.00201]