model category

for ∞-groupoids

Contents

Idea

The standard model category structure on cosimplicial objects in unital, commutative algebras over $k$.

Definition

Write $Alg_k^\Delta$ for the category of cosimplicial objects in the category of unital, commutative $k$-algebras. Sending algebras to their underlying $k$-modules yields a forgetful functor

$U : Alg_k^\Delta \to k Mod^\Delta \,.$

The model category structure

The Dold-Kan correspondence provides the normalized cochain complex (Moore complex) functor

$N : k Mod_k^\Delta \to Ch^\bullet(k)_+ \,.$

Define a morphism $f : A \to B$ of cosimplicial algebras is a morphism is a weak equivalence if

$N(U(f)) : N(U(A)) \to N(U(B))$

is a quasi-isomorphism in $Ch^\bullet_+(k)$.

Say a morphism of cosimplicial algebras is a fibration if it is a epimorphism (degreewise surjection).

This defines the projective model category structure on $Alg_k^\Delta$.

The simplicial model category structure

There is also the structure of an sSet-enriched category of $Alg_k^\Delta$.

Definition

For $X$ a simplicial set and $A \in Alg_k$ let $A^X \in Alg_k^\Delta$ be the corresponding $A$-valued cochains on simplicial sets

$A^X : [n] \mapsto \prod_{X_n} A_n \,.$
Remark

If we write $C(X) := Hom_{Set}(X_\bullet,k)$ for the cosimplicial algebra of cochains on simplicial sets then for $X$ degreewise finite this may be written as

$A^X = A \otimes C(X)$

where the tensor product is the degreewise tensor product of $k$-algebras.

Definition

For $A,B \in Alg_k^\Delta$ define the sSet-hom-object $Alg_k^\Delta(A,B)$ by

$Alg_k^\Delta(A,B) := Hom_{sSet}(A, B^{\Delta[\bullet]}) = Hom_{sSet}(A, B \otimes C(\Delta[\bullet])) \in sSet \,.$
Remark

For $B \in Alg_k$ regarded as a constant cosimplicial object under the canonical embedding $Alg_k \hookrightarrow Alg_k^\Delta$ we have

$Alg_k^\Delta(A, B^{\Delta[n]}) = Alg_k^\Delta(A, B \otimes C(\Delta[n])) \simeq Alg_k(A_n,B) \,.$
Proof

Let $f : A \to B \otimes C(\Delta[n])$ be a morphism of cosimplicial algebras and write

$f_n : A_n \to B$

for the component of $f$ in degree $n$ with values in the copy $B = B \otimes k$ of functions $k$ on the unique non-degenerate $n$-simplex of $\Delta[n]$. The $n+1$ coface maps $C(\Delta[n])_n \leftarrow C(\Delta[n])_{n-1}$ obtained as the pullback of the $(n+1)$ face inclusions $\Delta[n-1] \to \Delta[n]$ restrict on the non-degenerate $(n-1)$-cells to the $n+1$ projections $k \leftarrow k^{n+1} : p_i$.

Accordingly, from the naturality squares for $f$

$\array{ A_n &\stackrel{f_n}{\to}& B \\ \uparrow^{\mathrlap{\delta_i}} && \uparrow^{\mathrlap{p_i}} \\ A_{n-1} &\stackrel{f_{n-1}}{\to}& B^{n+1} }$

the bottom horizontal morphism is fixed to have components

$f_{n-1} = (f_n \circ \delta_0, \cdots, f_n \circ \delta_n)$

in the functions on the non-degenerate simplices.

By analogous reasoning this fixes all the components of $f$ in all lower degrees with values in the functions on degenerate simplices.

The above sSet-enrichment makes $Alg_k^\Delta$ into a simplicially enriched category which is tensored and cotensored over $sSet$.

And this is compatible with the model category structure:

Theorem

With the definitions as above, $Alg_k^\Delta$ is a simplicial model category.

Proof

To06, theorem 2.1.2

Under the monoidal Dold-Kan correspondence this is related to the model structure on commutative non-negative cochain dg-algebras.

References

Details are in section 2.1 of

The generalization to arbitrary cosimplicial rings is proposition 9.2 of

• J.L. Castiglioni G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence (arXiv)

There also aspects of relation to the model structure on dg-algebras is discussed. (See monoidal Dold-Kan correspondence for more on this).

Revised on September 18, 2014 09:43:04 by Urs Schreiber (185.26.182.29)