Question

The following question is related to my previous post on co-$A_\infty$ spaces (http://mathoverflow.net/questions/50298/co-a-infty-spaces), but goes in a somewhat different direction.

Some Background:

In trying to classify $A_\infty$ structures on a space $X$, one is led to obstructions living in Hochschild cohomology. One way to get this (I think), is to look at the tower

$$ \cdots \to A_n\text{-spaces} \to A_{n-1}\text{-spaces} \to \cdots \to A_2\text{-spaces,} $$

and one notes that the $n$-th layer is given by $\Omega^{n-2} F(X^{[n]},X)$, which is the $(n-2)$-fold loop space of the function space of based maps from the $n$-fold smash product of $X$ to $X$. Then the $k$-invariants (aka the maps inducing the $d^1$-differential in the homotopy spectral sequence) $$ \Omega^{n-2} F(X^{[n-1]},X) \to \Omega^{n-2} F(X^{[n]},X), \qquad n \ge 2 $$ can be computed explicitly and the formula for these is reminiscent of the Hochschild cohomology differential. More, precisely, if $X = \Omega Y$, and we look at stabilized versions of these function spaces, what I think one gets is the differential for topological Hochschild cohomology of the "group ring" $S[\Omega Y]$ (where $S =$ sphere spectrum; please correct me if I'm bungling this).

My Question:

What is the algebraic structure that arises when one tries to do deformation theory of co-$A_\infty$ (or suspension) structures on a space?

In this instance one has a tower as above, with "$A_n$" replaced by "co-$A_n$" at the $n$-th stage. But now the $k$-invariants in this case (at least in the stable range) are maps of spectra of the form: $$ \Omega^{n-2} F(X,W_{n-1}\wedge X^{[n-1]}) \to \Omega^{n-2} F(X,W_n \wedge X^{[n]}) $$ where $W_n$ is $(n-1)!$-copies of the $(1-n)$-sphere spectrum (yes, this is related to the Goodwillie tower of the identity functor).

So, my question amounts to the following:

What is the algebraic structure associated with this $k$-invariant? Is it some kind of "co-Hochschild" theory (whatever that means) of co-algebras? (where the co-algebra in this case is $X = \Sigma Y$).

Answer 1

As far as a Hochschild like differential there are the differentials `in the bar construction' p.44+ in LNM 161: H-spaces from a homotopy point of view and on p.54 description of the k-invariants of an A_n-space

Answer 2

You wrote: the tower

⋯→An-spaces→An−1-spaces→⋯→A2-spaces,

I'm not sure how to interpret that A_n-spaces means the cat of A_n-spaces?? OR A_n-space means X together with the structure maps OR A_n-space X<--> XP(n), the projective space but then XP(n)\subset XP(n+1) ??? but for none of these do I see Ωn−2F(X[n],X)???

what am I missing?

and one notes that the n-th layer is given by Ωn−2F(X[n],X), which is the (n−2)-fold loop space of the function space of based maps from the n-fold smash product of X to X. Then the k-invariants (aka the maps inducing the d1-differential in the homotopy spectral sequence) Ωn−2F(X[n−1],X)→Ωn−2F(X[n],X),n≥2 can be computed explicitly and the formula for these is reminiscent of the Hochschild cohomology differential.