nLab model structure on cosimplicial rings

Idea

A model category structure on cosimplicial objects in unital, commutative algebras over some field $k$.

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

The model structure

Let $k$ be a field of characteristic zero.

Definition

Write $cAlg_k^\Delta$ for the category of cosimplicial objects in the category of unital, commutative algebras over $k$.

Remark

Sending $k$-algebras to their underlying $k$-modules yields a forgetful functor

$$U \colon cAlg_k^\Delta \longrightarrow k Mod^\Delta$$

from cosimplicial $k$-allgebras (def. ) to cosimplicial objects in $k$-vector spaces.

Moreover, the Dold-Kan correspondence provides the normalized cochain complex functor

$$N \colon k Mod_k^\Delta \to Ch^{\geq 0}(k)$$

from cosimplicial $k$-vector spaces to cochain complexes (i.e. with differential of degree +1) in non-negative degrees.

Proposition

Say that morphism $f \colon A \to B$ in $cAlg_k^{\Delta}$ (def. ) is

1. a weak equivalence if its image $N(U(f)) \colon N(U(A)) \to N(U(B))$ under the comparison functors from remark is a quasi-isomorphism in $Ch^{\geq 0}(k)$;

1.a fibration if $f$ is an epimorphism (i.e. degreewise a surjection).

Then

1. this defines a model category structure, to be called the projective model structure on comsimplicial commutative $k$-algebras. $(cAlg_k^\Delta)_{poj}$.

2. this is a cofibrantly generated model category

3. and a simplicial model category.

e.g. Toën 00, theorem 2.1.2

Proof

The first two statements follow by observing that $(cAlg_k^{\Delta})_{proj} is$the transferred model structure along the forgetful functor $U \circ N$ from remark of the projective model structure on chain complexes, by this prop..

The third statement is the content of prop. below.

Properties

Simplicial model category structure

There is also the structure of an sSet-enriched category on $cAlg_k^\Delta$ (def. )

Definition

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

$$A^X \;\colon\; [n] \mapsto (A_n)^{X_n} = \underset{X_n}{\prod} A_n \,.$$

Remark

If we write $C(X) \coloneqq Hom_{Set}(X_\bullet,k)$ 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.

See also Castiglioni-Cortinas 03, p. 10.

Definition

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

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

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 $cAlg_k^\Delta$ into a simplicially enriched category which is tensored and cotensored over $sSet$.

And this is compatible with the model category structure:

Proposition

With the definitions as above, $(cAlg_k^\Delta)_{proj}$ is a simplicial model category.

Toën 00, theorem 2.1.2

Relation to the model structure on cochain dgc-algebras

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

References

Details are in

• Bertrand Toën, section 2.1 of Affine stacks (Champs affines) (arXiv:math/0012219) .

See also

• Paul Goerss, Rick 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:math/0306289)

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