Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n = 2,3,\dots$ that are inductively described (the definition of $c_n$ uses $c_{n-1}$ as input; the map $c_2$ is a co-$H$ structure). The suspension of a based space $X$ has the structure of a co-$A_\infty$ space.

Assume $Y$ is $2$-connected and has the homotopy type of a finite complex. Then Schwaenzl, Vogt and I showed that a co-$A_\infty$ space $Y$ desuspends to a space $X$ in the sense that there's a weak equivalence $\Sigma X \simeq Y$.

However we didn't try to check that the given weak equivalence is compatible in the co-$A_\infty$ sense. Part of the problem is that a morphism $f: Y \to Z$ of co-$A_\infty$ spaces should amount a co-$A_\infty$-structure on its mapping cylinder restricting to the given ones on $X \times 1$ and $Y$. However, this doesn't form a category: it's an $\infty$-category.

Now to my questions:

Question 1: is there a documented proof somewhere that the functor which assigns to a based space $X$ its suspension (considered as an co-$A_\infty$ space) induces an equivalence between the homotopy category of $1$-connected spaces and $2$-connected co-$A_\infty$ spaces?

Presumably, such a proof should be Hilton-Eckmann dual to one of the main results in the Book of Boardman and Vogt.

Question 2: Do function spaces coincide up to weak equivalence under this functor? That is, is the map $$\hom_{\text{Top}_*}(X,X') \to \hom_{\text{co-}A_\infty}(\Sigma X,\Sigma X')$$ A weak equivalence under suitable hypotheses on $X$ and $X'$?

By $\hom$ in each case, I mean topologized mapping spaces.

How would one go about proving a result like this?

share|improve this question
1  
Prof. Klein, there was some discussion of questions related to this here (but in a number of respects it seems you are already more informed): mathoverflow.net/questions/4117/… Also, welcome! –  Tyler Lawson Dec 24 '10 at 21:00
1  
Thanks Tyler, I wasn't aware of that discussion (and you can call me John if you wish). The two references that speak about matters in this direction are: 1. Hopkins, M.J. Formulations of cocategory and the iterated suspension. Algebraic homotopy and local algebra (Luminy, 1982), 212–226, Astérisque, 113-114, Soc. Math. France, Paris, 1984 2. Klein, J.; Schwänzl, R.; Vogt, R. M. Comultiplication and suspension. Topology Appl. 77 (1997), no. 1, 1–18 The first of these gives a Segal-type approach to desuspension (by finding a model for a cobar construction) which isn't operadic. –  John Klein Dec 24 '10 at 21:33
add comment

1 Answer

I have the feeling many of us would agree those statements `should' be true, but I can't think of anyone who wrote things down publicly. Why not ask at alg-top? overflow requires active logging in alg-top doesn't

jim

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.