model structure on cosimplicial abelian groups



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Let Ab ΔAb^{\Delta} be the category of cosimplicial objects in the category Ab of abelian groups – the category of cosimplicial abelian groups .

This entry discusses structures of model categories on Ab ΔAb^\Delta.

By the dual Dold-Kan correspondence there is an equivalence of categories Ab ΔNΞCh + (Ab)Ab^\Delta \stackrel{\overset{\Xi}{\leftarrow}}{\underset{N}{\to}} Ch^\bullet_+(Ab) with the category of cochain complexes in non-negative degree. Since Ab is an abelian category, we have by general results various model structures on cochain complexes. Via the Dold-Kan equivalence, all of these induce model structures on Ab ΔAb^\Delta.


Simplicial enrichment of the projective structure

Since Ab has all limits and colimits, the category of cosimplicial objects (as described there) Ab ΔAb^\Delta inherits canonically the structure of an sSet-enriched category which is powered and copowered.

Write Ab proj ΔAb^\Delta_{proj} for the model structure that is induced by the dual Dold-Kan correspondence Ab ΔCh + (Ab)Ab^\Delta \simeq Ch^\bullet_+(Ab) from the model structure on cochain complexes whose fibrations are the degreewise surjections (and weak equivalences the usual quasi-isomorphisms). This is described in detail here. So this induces the model structure Ab proj ΔAb^\Delta_{proj} whose fibrations are also the degreewise surjections in AbAb (using that the normalized cochain complex-functor preserves surjections.)


The canonical sSetsSet-enrichement of Ab ΔAb^\Delta is compatible with the model category structure Ab proj ΔAb^\Delta_{proj} in that the combination gives Ab ΔAb^\Delta the structure of a simplicial model category.


We need check the pushout-product axiom of an enriched model category of the standard model structure on simplicial sets sSet QuillensSet_{Quillen}

So we need to show that for i:CCi : C \to C' any cofibration in sSet QuillensSet_{Quillen} and j:XYj : X \to Y a fibration of cosimplcial abelian groups (degreewise surjection) the morphism

k:X CX C× Y CY C k : X^{C'} \to X^C \times_{Y^C} Y^{C'}

induced by the powering () ():Ab Δ×sSetAb Δ(-)^{(-)} : Ab^\Delta \times sSet \to Ab^\Delta is a fibration, which is acyclic if ii or jj is.

That kk is a fibration is easily checked. To see acyclicity we first notice the following

Lemma. If i:CCi : C \to C' is a weak equivalence then for every cosimplicial abelian group AA we have A iA^i is a weak equivalence.

To see this observe that A CA^C is the diagonal of an evident bisimplicial abelian group and that A iA^i is then in one argument a degreewise quasi-isomorphism. Since forming total complexes preserves degreewise equivalences, the lemma follows.

To continue the main proof, notice that we have a short exact sequence

0X C× Y CY CX CY CfY C0 0 \to X^C \times_{Y^C} Y^{C'} \to X^C \oplus Y^{C'} \stackrel{f}{\to} Y^C \to 0

with f:(x,y)j C(x)Y i(y)f : (x,y) \mapsto j^C(x) - Y^i(y). This induces a long exact sequence in cohomology

H p(X C× Y CY C)H p(X C)H i(Y C)H p(Y C)H p+1(X C× Y CY C). \cdots \to H^p(X^C \times_{Y^C} Y^{C'}) \to H^p(X^C) \oplus H^i(Y^{C'}) \to H^p(Y^C) \to H^{p+1}(X^C \times_{Y^C} Y^{C'}) \to \cdots \,.

If ii is a weak equivalence, then by the above lemma we have that

ker(H p(X C)H p(Y C)H p(Y C))H p(X C). ker(H^p(X^C) \oplus H^p(Y^{C'}) \to H^p(Y^C)) \simeq H^p(X^C) \,.

Inspection of the connecting homomorphism then shows that H p(Y C)H p+1(X C× Y CY C)H^p(Y^C) \to H^{p+1}(X^C \times_{Y^C} Y^{C'}) is the 0-map. In total this implies that we have an isomorphism

H p(X C× Y CY C)H p(X C) H^p(X^C \times_{Y^C} Y^{C'}) \stackrel{\simeq}{\to} H^p(X^C)

for all pp, and hence that

X C× Y CY CX C X^C \times_{Y^C} Y^{C'} \to X^C

is a weak equivalence. Since by the above lemma also X i:X CX CX^i : X^{C'} \to X^C is a weak equivalence, it follows by 2-out-of-3 that the morphism kk is indeed a weak equivalence if ii is.

An analogous argument shows that kk is a weak equivalence if jj is.

This argument is essentially that on page 41 of (Toën)


The model structure on cosimplicial algebras is discussed in detail in

The above proof that Ab proj ΔAb^\Delta_{proj} is a simplicial model category mimics the proof on page 41 there. Indeed, the claim is that the model structure on cosimplicial algebras is the transferred model structure induced by the above from the evident forgetful functor.

Last revised on July 14, 2021 at 06:56:43. See the history of this page for a list of all contributions to it.