# Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's $\Gamma$ functor on groups, a $\ast$-product on the bar complex (which might be essentially the same thing as Loday's filtration of Hochschild homology by shuffle products, in a different context), iterated normalized bar constructions, and a whole slew of different products.

There is a notion of Eilenberg-MacLane stacks for the additive group $\mathbb{G}_a$ (and presumably for other group schemes?), described in Toen's Champs Affines. Has the same sort of analysis been carried out for such "derived" Eilenberg MacLane spaces? That is, is it known that such stacks can be constructed by some sensible iterated bar construction? Do they have the same sorts of products on them? What exactly does homology of such simplicial schemes compute, stack (co)homology?

There is for instance an interesting result of Eilenberg and MacLane (26.3 in paper II of On the Groups $H(\Pi,n)$) indicating that $H^3(K(\Pi,1),K(\Pi,1)^\ast;G)=0$, where $K(\Pi,1)^\ast$ is the subcomplex generated by the shuffle product of chains (they actually use the complexes $A(\Pi,1)$, which have identical homology but are "more perspicuous"). The proof seems to be completed by suspending chains in $K(\Pi,1)$ and looking at the cohomology of $K(\Pi,2)$. This seems that it would be even more interesting in the context of group schemes, since the double delooping of $\mathbb{G}_a$ is necessarily a stack. The cohomology of such stacks also seems to be the natural place to work with the so-called "polynomial cocycles" of Robert Heaton and others, which behave like group cohomology but don't actually make sense as cocycles in group cohomology.

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May be can help watch: "books.google.it/books/about/…; ANd see the j.Duskin work on: tac.mta.ca/tac/reprints/articles/23/tr23abs.html –  Buschi Sergio Jul 27 '13 at 19:59
What do you mean by "good notion" with respect to Eilenberg-MacLane stacks? A solid criterion seems to be what we need when deciding whether iterated bar constructions for other commutative group schemes/sheaves/stacks yield "good" objects. –  S. Carnahan Jul 27 '13 at 22:32
I don't mean anything. I'll change it to just say "a notion." It seemed pretty good to me. –  Jon Beardsley Jul 28 '13 at 3:34
I'm working on writing down some of my own analysis of such things, for the moment. –  Jon Beardsley Jul 29 '13 at 18:07