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. ) 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. ) 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 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 of the projective model structure on chain complexes, by this prop..

The third statement is the content of prop. below.


Simplicial model category structure

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


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.