# cosimplicial algebras to dg-algebras

The normalized Moore complex functor is usually considered taking simplicial abelian groups to chain complexes. But there is an obvious dual version that takes cosimplicial abelian groups to N-graded cochain complexes.

Moreover, applied on a cosimplicial group that has the structure of a cosimplicial ring, the Moore cochain complex yields a cochain complex that has the structure of a dg-algebras (the details are at monoidal Dold-Kan correspondence).

This seems obvious and useful enough, but I find surprisingly little literature on this dual monoidal Dold-Kan correspondence. In fact the only relevant reference that I am aware of is Castiglioni-Cortinas, Cosimplicial versus dg-rings: a version of the Dold-Kan correspondence.

They consider not the Moore cochain functor but its right adjoint, and show that its left derived functor is an equivalence of homotopy categories.

But it would seem that instead considering directly the Moore cochain complex functor on cosimplicial rings would be at least as interesting.

I can dream up some of its properties myself, but I keep feeling I must be missing the canonical literature on this, which must exist. Does anyone have more references on the Moore cochain complex functor on cosimplicial rings/algebras?

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Hooray for lots of background links! +1 –  Scott Morrison Oct 27 '09 at 1:21

This construction is closely related to de Rham cohomology, and used to compute the continuous cohomology of Lie groups. This is part of the tools to compute Beilinson's regulator, as explained in

J. I. Burgos Gil, The regulators of Beilinson and Borel. CRM Monograph Series, 15. American Mathematical Society, Providence, RI, 2002.

See Chapter 7 for the abstract results, and Chapter 8 for its applications. (It seems that there is an online version here.)

The same kind of constructions is considered by Larry Breen and Bill Messing in

Combinatorial differential forms, Advances in Mathematics, vol. 164 (2001), 203-282

and used in this paper of Larry Breen arXiv:0802.1833 to study differential geometry on gerbes ("non-abelian de Rham cohomology").

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What you list are indeed pretty much the examples that motivate the question. I talk about these examples in the example section here: ncatlab.org/schreiber/show/Chevalley-Eilenberg+algebra But what I have not seen in the literature much is a discussion of the general abstract property of this functor from cosimplicial algebras to dg-algebras. Thanks for the reference by Gil. I hadn't seen that before. Will have a look. –  Urs Schreiber Nov 18 '09 at 18:12
Burgos's book contains some abstract results: you might have a look at theorem 7.3 in Burgos's book (due to Beilinson), which gives a rather nice equivalence of categories between reduced dg algebras and "small" cosimplicial algebras. –  Denis-Charles Cisinski Nov 18 '09 at 18:58
Ah, thanks. Yes, that section 7 is useful. Thanks. That smallness assumption in def 7.2 applies to some cases of interest, but is pretty restrictive. There should be a more general statement like this: while the dg-algebra NX corresponding to a cosimplicial algebra X need not be commutative, it is always an E-oo-algebra! So the full statement here should probably be that Dold-Kan induces an oo-equivalence of E-oo algebra objects in cosimplicial abelian groups and E-oo algebra objects in N-graded cochain complexes. Have you seen anything like this stated? –  Urs Schreiber Nov 18 '09 at 19:51
For clarification: I am talking about the E-oo structure that is well known for cochains on simplicial sets (as here ncatlab.org/nlab/show/…) . This should immediately generalize to all dg-algebras coming from cosimplicial algebras. Not just cosimplicial algebras of functions on a simplicial set. –  Urs Schreiber Nov 18 '09 at 20:07