Cosimplicial Objects and little n-cubes. I

Abstract

In this paper we show that if a cosimplicial space or spectrum $X^{\bullet }$ has a certain kind of combinatorial structure (we call it a ${\Xi }^n$-structure) then the total space of $X^\b$ has an action of a certain operad which is weakly equivalent to the little n-cubes operad. The $n\le 2$ case was proved by a more complicated argument in our earlier paper A Solution of Deligne's Hochschild Cohomology Conjecture (http://front.math.ucdavis.edu/math.QA/9910126). In the special case $n=\infty$, we define a symmetric monoidal structure $⊠$ on cosimplicial spaces and show that if $X^\b$ is a commutative $⊠$-monoid then the total space of $\X^\b$ is an $E_{\infty }$ space.

There are three new sections: Section 10 shows that ${\Xi }^2$-structures are essentially the same thing as operads with multiplication, Section 11 shows that the operad ${\mathcal{D}}_n$ acts on $n$-fold loop spaces, and Section 15 shows that the main results are still valid for the homotopy-invariant version of Tot