Nikita Markarian’s mathblog

Classifying operad

Posted in Mathematics by nikitamarkarian on June 5, 2009

In the previous post everything is based on the following observation. Let A be an algebra over chains of small discs operad. Then A is in particular a homotopy algebra. I propose that on Hochschild homology of A there is a natural action of an operad equivalent to the
associative operad Ass (this is essentially proved in M. Brun, Z. Fiedorowicz, R. Vogt “On the multiplicative structure of topological Hochschild homology”). More generally, one may try introduce of action (whatever it means) of an operad equivalent to Ass on the category of twisted (semifree) complexes of A and derive the previous action from this one.

Thinking about a proof of this statement I formulated a more general question.
(Below I easily pass from things (operads) to homotopy equivalent ones and ignore the fact that \Omega BX\neq X in general.)

Let us start with the category of topological spaces. Let C be an operad  and M be the corresponding monad: M(X)=\amalg C(n) \times X^n. Suppose that a morphism \epsilon\colon\Omega S\to C is given. It follows that any C-space X is a H-space.  One may take the its  classifying space BX. Question: if X is a C-space, what operad does naturally act on BX? Call this operad (if any) by the classifying operad B(C).  The monad B(M) corresponding to the classsifying  should solve equation \Omega B(M) S = M. Indeed, if X is a space over B(M), then M naturally acts on \Omega X: M\Omega X= \Omega B(M) S \Omega X \to \Omega B(M) X \to \Omega X, where the first arrow is the canonical morphism S \Omega \to id and the second one is given by B(M)-action on X. The solution may be easily found. This is a simplicial monad, its components are S\Omega S\dots \Omega S M \Omega S\dots S\Omega and structure morphisms are given by the canonical  morphism S\Omega\to id and morphisms \Omega S M\to M and M \Omega S that are compositions of \epsilon\colon\Omega S\to M with the multiplication M M\to M. This is obviously a version of construction from chapter 9 of the May’s book. Classifying space may be represented by simplicial space S\Omega S\dots \Omega S X. One may easily write down the B(M)-action on it.

One may ask if there exist a more economical construction of B(M) by analogy with classical construction of classifying space.

The monad B(M) is not an operad in general, because S is not invertible, but it is an operad in the stable homotopy category. But if we are lucky, it is. For example, classifying operad of E_n (the thing equivalent to \Omega^n S^n) is E_{n-1}.

From the ancient times there are two approaches to delooping, E_n-spaces etc.: of May and of Boardman-Vogt (see Adams, Infinite loop spaces \S 2.4). An important (and mysterious) notion in Boardman-Vogt approach is the tensor product of operads, in particular product with Ass, operad of associative algebras (see  J. M. Boardman R. M.  Vogt “Homotopy Invariant Algebraic Structures On Topological Spaces” or the mentioned paper and references therein). The classifying operad must correspond to it in the May approach. But it more division by Ass rather than multiplication. That is it is natural to suppose that B(C\otimes Ass)=C.

Now let us pass from the category of topological space to category of complexes (taking homology) and try to mimic the construction discussed. So we start with an operad C in the category of complexes with a morphism Ass \to C; algebra over this operad is an algebra proper, in particular. The definition of classifying operad works on the level of homology, i.e. it works for operads in the category of complexes. Example: the classifying operad of  chains of small discs operad is homotopy equivalent to  operad Ass. Another example: the classifying operad of the Gerstenhaber operad is the operad acting on a Poisson algebra. As the Poisson operad is different from Ass, it proves that, besides the Gerstenhaber operad is homotopy equivalent to  chains of small discs operad (by the Tamarkin’s theorem), the equivalence does not respect natural maps from Ass to these operads.

The role of the classifying operad in the category of complexes is not so obvious. That is where does it naturally acts? (Note, that to mimic the definition of “classifying space” of an algebra one need an augmentation map: the “homology of the classifying space” is the homology of the algebra in the augmentation module; it gives an answer on the the question posed for the category of algebras with augmentation.)

I suggest two conjectural solution. Firstly given an algebra (proper) on may organise the category X of semi-free modules over it (or category of twisted complexes for A_\infty-algebras). If the algebra is an algebra over an operad C, then the category has additional structure: operad B(C) acts on it. The latter is not very clear. The only way I can interpret it is as a morphism from C(n) to homology of nerve of category of functors from X^n to X such, that… Example: category of modules over a 2-algebra (algebra over chains of the small disks operad) is a (homotopy) tensor category.

The second more promising suggestion is to define an action of B(C) on the Hochschild homology of an algebra. Indeed, on the level of topological spaces, Hochschild homology corresponds to taking of free loop space of the classifying space. As the classifying operad  acts on the classifying space it acts on its free loop space as well.   On Hochschild homology Rinehart-Connes-Feigin-Tsygan-Loday-Quillen differential also acts. It would be  extremely interesting to understand how it interacts with the classifying operad. Example: Hochschild homology of a 2-algebra is an algebra (see the mentioned paper).

Finally let me note that it seems natural to start with a PROP. That is given a PROP with a morphism from \Omega S to it and a space over it. What PROP (or, at least operad) does  naturally act on the classifying space of this space? Unfortunately, I know nothing about PROPS.

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