nLab
simplicial bar construction

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Limits and colimits

Contents

Idea

The simplicial bar construction is a way of building a cofibrant resolution of a diagram, suitable for computing homotopy colimits in simplicial model categories.

Definition

Let 𝒱\mathcal{V} be a cocomplete symmetric monoidal closed category and let \mathbb{C} be a small 𝒱\mathcal{V}-category. We define a simplicial object W (c,,c)W_\bullet (c', \mathbb{C}, c) for each pair (c,c)(c', c) of objects in \mathbb{C} by

W n(c,,c)= (c 0,,c n)(c n,c)(c n1,c n)(c 0,c 1)(c,c 0)W_n (c', \mathbb{C}, c) = \coprod_{(c_0, \ldots, c_n)} \mathbb{C} (c_n, c') \otimes \mathbb{C} (c_{n-1}, c_n) \otimes \cdots \otimes \mathbb{C} (c_0, c_1) \otimes \mathbb{C} (c, c_0)

with the obvious face and degeneracy operators induced by the composition and identities of \mathbb{C}. Note that W (blank,,blank)W_\bullet (\blank, \mathbb{C}, \blank) is a 𝒱\mathcal{V}-functor op[Δ op,𝒱]\mathbb{C}^{op} \otimes \mathbb{C} \to [\mathbf{\Delta}^{op}, \mathcal{V}].

Let \mathcal{M} be a tensored 𝒱\mathcal{V}-category. The two-sided simplicial bar construction of a 𝒱\mathcal{V}-diagram F:F : \mathbb{C} \to \mathcal{M} weighted by a 𝒱\mathcal{V}-functor G: op𝒱G : \mathbb{C}^{op} \to \mathcal{V} is a simplicial object B (G,,F)B_\bullet (G, \mathbb{C}, F) equipped with isomorphisms

(B n(G,,F),M) (c,c): op𝒱(W n(c,,c),(GcFc,M))\mathcal{M} (B_n (G, \mathbb{C}, F), M) \cong \int_{(c', c) : \mathbb{C}^{op} \otimes \mathbb{C}} \mathcal{V} (W_n (c', \mathbb{C}, c), \mathcal{M}(G c' \odot F c, M))

that are natural in nn and 𝒱\mathcal{V}-natural in MM, where the integral sign on the right hand side denotes a 𝒱\mathcal{V}-end.

Now suppose a cosimplicial object Δ :Δ𝒱\Delta^\bullet : \mathbf{\Delta} \to \mathcal{V} is given, so that we may define the realisation? of a simplicial object X X_\bullet in a 𝒱\mathcal{V}-category as the weighted colimit |X |=Δ X \left| X_\bullet \right| = \Delta^\bullet \star X_\bullet. The two-sided bar construction of F:F : \mathbb{C} \to \mathcal{M} weighted by G: op𝒱G : \mathbb{C}^{op} \to \mathcal{V} is then defined to be the geometric realisation of the two-sided simplicial bar construction:

B(G,,F)=|B (G,,F)|B (G, \mathbb{C}, F) = \left| B_\bullet (G, \mathbb{C}, F) \right|

Properties

Proposition

If \mathcal{M} is a tensored 𝒱\mathcal{V}-category and has coproducts for small families of objects, then \mathcal{M} has two-sided simplicial bar constructions for all diagrams and weights. If \mathcal{M} is moreover 𝒱\mathcal{V}-cocomplete, then \mathcal{M} also has two-sided bar constructions.

Proposition

If \mathcal{M} is a tensored 𝒱\mathcal{V}-category and has coproducts for small families of objects, then:

  • For each 𝒱\mathcal{V}-diagram F:F : \mathbb{C} \to \mathcal{M}, the 𝒱\mathcal{V}-functor B n(,,F):[ op,𝒱]B_n (-, \mathbb{C}, F) : [\mathbb{C}^{op}, \mathcal{V}] \to \mathcal{M} has a right 𝒱\mathcal{V}-adjoint.

If \mathcal{M} is also cotensored, then:

  • For each weight G: op𝒱G : \mathbb{C}^{op} \to \mathcal{V}, the 𝒱\mathcal{V}-functor B n(G,,):[,]B_n (G, \mathbb{C}, -) : [\mathbb{C}, \mathcal{M}] \to \mathcal{M} has a right 𝒱\mathcal{V}-adjoint.
Proposition

Let W(c,,c)W (c', \mathbb{C}, c) be the realisation of W (c,,c)W_\bullet (c', \mathbb{C}, c). If \mathcal{M} is a cotensored 𝒱\mathcal{V}-cocomplete 𝒱\mathcal{V}-category, then there are isomorphisms

(B(G,,F),M) (c,c): op𝒱(W(c,,c),(GcFc,M))\mathcal{M} (B (G, \mathbb{C}, F), M) \cong \int_{(c', c) : \mathbb{C}^{op} \otimes \mathbb{C}} \mathcal{V} (W (c', \mathbb{C}, c), \mathcal{M} (G c' \odot F c, M))

that are 𝒱\mathcal{V}-natural in MM.

The next theorem is essentially a version of the Fubini theorem for coends.

Theorem

Let F:F : \mathbb{C} \to \mathcal{M}, G:𝔻 op𝒱G : \mathbb{D}^{op} \to \mathcal{V}, and H: op×𝔻𝒱H : \mathbb{C}^{op} \times \mathbb{D} \to \mathcal{V} be 𝒱\mathcal{V}-functors. If \mathcal{M} is a cotensored 𝒱\mathcal{V}-cocomplete 𝒱\mathcal{V}-category, then we have the following isomorphism:

B(B(G,𝔻,H),,F)B(G,𝔻,B(H,,F))B (B (G, \mathbb{D}, H), \mathbb{C}, F) \cong B (G, \mathbb{D}, B (H, \mathbb{C}, F))

Applications

As stated in the introduction, the two-sided simplicial bar construction can be used to compute homotopy colimits.

Let us write ycy c for the representable 𝒱\mathcal{V}-functor (,c)\mathbb{C}(-, c) and B(,,F)B (\mathbb{C}, \mathbb{C}, F) for the diagram cB(yc,,F)c \mapsto B (y c, \mathbb{C}, F).

Theorem

Let \mathcal{M} be a simplicial model category and let \mathbb{C} be a small category.

  1. For each diagram F:F : \mathbb{C} \to \mathcal{M}, there is a weak equivalence B(yc,,F)FcB (y c, \mathbb{C}, F) \to F c that is natural in FF and cc.
  2. The functor colim:[,]colim : [\mathbb{C}, \mathcal{M}] \to \mathcal{M} preserves weak equivalences between diagrams of the form B(,,F)B (\mathbb{C}, \mathbb{C}, F) where FF is a diagram such that each FcF c is a cofibrant object in \mathcal{M}.
  3. The functor B(1,,Q):[,]B (1, \mathbb{C}, Q -) : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}, where Q:Q : \mathcal{M} \to \mathcal{M} is a cofibrant replacement functor, is a left derived functor for colim:[,]colim : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}, i.e. it computes homotopy colimits.

References

Last revised on August 3, 2013 at 23:56:05. See the history of this page for a list of all contributions to it.