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I am thinking about two duality theorems for H-spaces and their actions.

By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider the actions of an H-space, which produce monoid actions in the derived (homotopy) category. We can also define $- \wedge_A -$ and [-,-]${}_A$ for an H-space $A$.

Let A be an H space and let X be an A-action. I am interested in these basic conditions involving duality in this setting:

  • Can you characterize when there is a weak equivalence $[[X,A],A] \cong X$ (respectively, homotopy equivalence).
  • Can you characterize when there is a weak equivalence $[X,Y] \cong [X,A] \wedge Y$?
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    $\begingroup$ Actions in the homotopy category are too incoherent to define $-\wedge_A-$ and $[-,-]_A$. These constructions only make sense for actions of $E_1$-monoids in the $\infty$-category of spaces (they are defined by a simplicial colimit and cosimplicial limit, respectively), and $A$ should be at least $E_2$ if you want the output to still have an action of $A$. $\endgroup$
    – Marc Hoyois
    Commented Feb 11 at 19:09
  • $\begingroup$ @MarcHoyois do you mean that they aren't as extensive as E₁-monoids and so less natural, or rather that there is no definition? Because it seems like you can have taken a homotopy coequilizer of the two maps X ∧ A ∧ Y ⭢ X ∧Y, and a homotopy equalizer of the two maps A ∧ [X,Y] ⭢ [X,Y]. $\endgroup$
    – user30211
    Commented Feb 14 at 13:18
  • $\begingroup$ Homotopy colimit constructions are not defined in the homotopy category, that's why this only makes sense in the $\infty$-category. Even then, the correct definition of $\wedge_A$ is as the colimit of a whole simplicial diagram (the so-called bar construction), not just as a coequalizer. $\endgroup$
    – Marc Hoyois
    Commented Feb 15 at 10:54

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