homotopy theory, (∞,1)-category theory, homotopy type theory
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on chain complexes/model structure on cosimplicial abelian groups
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The simplicial bar construction is a way of building a cofibrant resolution of a diagram, suitable for computing homotopy colimits in simplicial model categories.
Let $\mathcal{V}$ be a cocomplete symmetric monoidal closed category and let $\mathbb{C}$ be a small $\mathcal{V}$-category. We define a simplicial object $W_\bullet (c', \mathbb{C}, c)$ for each pair $(c', c)$ of objects in $\mathbb{C}$ by
with the obvious face and degeneracy operators induced by the composition and identities of $\mathbb{C}$. Note that $W_\bullet (\blank, \mathbb{C}, \blank)$ is a $\mathcal{V}$-functor $\mathbb{C}^{op} \otimes \mathbb{C} \to [\mathbf{\Delta}^{op}, \mathcal{V}]$.
Let $\mathcal{M}$ be a tensored $\mathcal{V}$-category. The two-sided simplicial bar construction of a $\mathcal{V}$-diagram $F : \mathbb{C} \to \mathcal{M}$ weighted by a $\mathcal{V}$-functor $G : \mathbb{C}^{op} \to \mathcal{V}$ is a simplicial object $B_\bullet (G, \mathbb{C}, F)$ equipped with isomorphisms
that are natural in $n$ and $\mathcal{V}$-natural in $M$, where the integral sign on the right hand side denotes a $\mathcal{V}$-end.
Now suppose a cosimplicial object $\Delta^\bullet : \mathbf{\Delta} \to \mathcal{V}$ is given, so that we may define the realisation? of a simplicial object $X_\bullet$ in a $\mathcal{V}$-category as the weighted colimit $\left| X_\bullet \right| = \Delta^\bullet \star X_\bullet$. The two-sided bar construction of $F : \mathbb{C} \to \mathcal{M}$ weighted by $G : \mathbb{C}^{op} \to \mathcal{V}$ is then defined to be the geometric realisation of the two-sided simplicial bar construction:
If $\mathcal{M}$ is a tensored $\mathcal{V}$-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 $\mathcal{V}$-cocomplete, then $\mathcal{M}$ also has two-sided bar constructions.
If $\mathcal{M}$ is a tensored $\mathcal{V}$-category and has coproducts for small families of objects, then:
If $\mathcal{M}$ is also cotensored, then:
Let $W (c', \mathbb{C}, c)$ be the realisation of $W_\bullet (c', \mathbb{C}, c)$. If $\mathcal{M}$ is a cotensored $\mathcal{V}$-cocomplete $\mathcal{V}$-category, then there are isomorphisms
that are $\mathcal{V}$-natural in $M$.
The next theorem is essentially a version of the Fubini theorem for coends.
Let $F : \mathbb{C} \to \mathcal{M}$, $G : \mathbb{D}^{op} \to \mathcal{V}$, and $H : \mathbb{C}^{op} \times \mathbb{D} \to \mathcal{V}$ be $\mathcal{V}$-functors. If $\mathcal{M}$ is a cotensored $\mathcal{V}$-cocomplete $\mathcal{V}$-category, then we have the following isomorphism:
As stated in the introduction, the two-sided simplicial bar construction can be used to compute homotopy colimits.
Let us write $y c$ for the representable $\mathcal{V}$-functor $\mathbb{C}(-, c)$ and $B (\mathbb{C}, \mathbb{C}, F)$ for the diagram $c \mapsto B (y c, \mathbb{C}, F)$.
Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category.
Last revised on August 3, 2013 at 23:56:05. See the history of this page for a list of all contributions to it.