model structure on cosimplicial rings


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

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 kk be a field of characteristic zero.


Write cAlg k ΔcAlg_k^\Delta for the category of cosimplicial objects in the category of unital, commutative algebras over kk.


Sending kk-algebras to their underlying kk-modules yields a forgetful functor

U:cAlg k ΔkMod Δ U \colon cAlg_k^\Delta \longrightarrow k Mod^\Delta

from cosimplicial kk-allgebras (def. 1) to cosimplicial objects in kk-vector spaces.

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

N:kMod k ΔCh 0(k) N \colon k Mod_k^\Delta \to Ch^{\geq 0}(k)

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


Say that morphism f:ABf \colon A \to B in cAlg k ΔcAlg_k^{\Delta} (def. 1) is

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

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


  1. this defines a model category structure, to be called the projective model structure on comsimplicial commutative kk-algebras. (cAlg k Δ) poj(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


The first two statements follow by observing that (cAlg k Δ) projis(cAlg_k^{\Delta})_{proj} isthe transferred model structure along the forgetful functor UNU \circ N from remark 1 of the projective model structure on chain complexes, by this prop..

The third statement is the content of prop. 2 below.


Simplicial model category structure

There is also the structure of an sSet-enriched category on cAlg k ΔcAlg_k^\Delta (def. 2)


For XX a simplicial set and AAlg kA \in Alg_k let A XAlg k ΔA^X \in Alg_k^\Delta be the corresponding AA-valued cochains on simplicial sets

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

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

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

where the tensor product is the degreewise tensor product of kk-algebras.

See also Castiglioni-Cortinas 03, p. 10.


For A,BAlg k ΔA,B \in Alg_k^\Delta define the sSet-hom-object Alg k Δ(A,B)Alg_k^\Delta(A,B) by

Alg k Δ(A,B)Hom sSet(A,B Δ[])=Hom sSet(A,BC(Δ[]))sSet. Alg_k^\Delta(A,B) \coloneqq Hom_{sSet}(A, B^{\Delta[\bullet]}) = Hom_{sSet}(A, B \otimes C(\Delta[\bullet])) \in sSet \,.

For BAlg kB \in Alg_k regarded as a constant cosimplicial object under the canonical embedding Alg kAlg k ΔAlg_k \hookrightarrow Alg_k^\Delta we have

Alg k Δ(A,B Δ[n])=Alg k Δ(A,BC(Δ[n]))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) \,.

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

f n:A nB f_n : A_n \to B

for the component of ff in degree nn with values in the copy B=BkB = B \otimes k of functions kk on the unique non-degenerate nn-simplex of Δ[n]\Delta[n]. The n+1n+1 coface maps C(Δ[n]) nC(Δ[n]) n1C(\Delta[n])_n \leftarrow C(\Delta[n])_{n-1} obtained as the pullback of the (n+1)(n+1) face inclusions Δ[n1]Δ[n]\Delta[n-1] \to \Delta[n] restrict on the non-degenerate (n1)(n-1)-cells to the n+1n+1 projections kk n+1:p ik \leftarrow k^{n+1} : p_i.

Accordingly, from the naturality squares for ff

A n f n B δ i p i A n1 f n1 B n+1 \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 n1=(f nδ 0,,f nδ n)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 ff in all lower degrees with values in the functions on degenerate simplices.

The above sSet-enrichment makes cAlg k ΔcAlg_k^\Delta into a simplicially enriched category which is tensored and cotensored over sSetsSet.

And this is compatible with the model category structure:


With the definitions as above, (cAlg k Δ) proj(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.


Details are in

See also

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).

Last revised on February 23, 2017 at 11:36:59. See the history of this page for a list of all contributions to it.