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model structure on cosimplicial rings

Idea

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

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

Definition

Write ${\mathrm{Alg}}_k^{\Delta }$ 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

$$U:{\mathrm{Alg}}_k^{\Delta }\to k{\mathrm{Mod}}^{\Delta }.$$U : Alg_k^\Delta \to k Mod^\Delta \,.

The model category structure

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

$$N:k{\mathrm{Mod}}_k^{\Delta }\to {\mathrm{Ch}}^{•}(k{)}_{+}.$$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

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

is a quasi-isomorphism in ${\mathrm{Ch}}_{+}^{•}(k)$.

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

This defines the projective model category structure on ${\mathrm{Alg}}_k^{\Delta }$.

The simplicial model category structure

There is also the structure of an sSet-enriched category of ${\mathrm{Alg}}_k^{\Delta }$.

Definition

For $X$ a simplicial set and $A\in {\mathrm{Alg}}_k$ let $A^X\in {\mathrm{Alg}}_k^{\Delta }$ be the corresponding $A$-valued cochains on simplicial sets

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

Remark

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

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

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

See also CasCor, p. 10.

Definition

For $A,B\in {\mathrm{Alg}}_k^{\Delta }$ define the sSet-hom-object ${\mathrm{Alg}}_k^{\Delta }(A,B)$ by

$${\mathrm{Alg}}_k^{\Delta }(A,B):={\mathrm{Hom}}_{\mathrm{sSet}}(A,B^{\Delta [•]})={\mathrm{Hom}}_{\mathrm{sSet}}(A,B\otimes C(\Delta [•]))\in \mathrm{sSet}.$$Alg_k^\Delta(A,B) := Hom_{sSet}(A, B^{\Delta[\bullet]}) = Hom_{sSet}(A, B \otimes C(\Delta[\bullet])) \in sSet \,.

Remark

For $B\in {\mathrm{Alg}}_k$ regarded as a constant cosimplicial object under the canonical embedding ${\mathrm{Alg}}_k\hookrightarrow {\mathrm{Alg}}_k^{\Delta }$ we have

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

$$f_n:A_n\to B$$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 $f$

$$\begin{array}{ccc}{A}_n& \stackrel{f_n}{\to }& B\\ {\uparrow }^{{\delta }_i}& & {\uparrow }^{p_i}\\ A_{n-1}& \stackrel{f_{n-1}}{\to }& B^{n+1}\end{array}$$\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 ${\mathrm{Alg}}_k^{\Delta }$ into a simplicially enriched category which is tensored and cotensored over $\mathrm{sSet}$.

And this is compatible with the model category structure:

Theorem

With the definitions as above, ${\mathrm{Alg}}_k^{\Delta }$ 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

• Bertrand Toen, Affine stacks (Champs affines) (arXiv:math/0012219) .

See also

• Goerrs, Jardine, Simplicial homotopy theory

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).