Double hom with $\mathbb{CP}^\infty$

Question

Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy.

$\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\infty}$, is an H-space, and an internal abelian group in the derived category. I am interested in algebras for the monad of $[[-,\textbf{B}\textbf{B}\mathbb{Z}], \textbf{B}\textbf{B}\mathbb{Z}]$, and producing a double-dual theorem of a similar nature.

It could go like this: let $[-,\textbf{B} \textbf{B}\mathbb{Z}]\text{-alg}$ be the category of algebras for $[[-,\textbf{B} \textbf{B}\mathbb{Z}],\textbf{B} \textbf{B}\mathbb{Z}]$. I am looking to endow this with an internal hom $[-,-]_{[-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg}}$ and then to demonstrate a categorial equivalence like so:

$$[-,\textbf{B} \textbf{B}\mathbb{Z}]_{[-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg}} : [-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg} \leftrightarrow \text{H-Spc} : [-,\textbf{B}\textbf{B}\mathbb{Z}]_{\text{H-Spc}}$$

In the above, $\text{H-Spc}$ is the homotopy category of H-spaces.

Some ideas:

1. Maybe only the H-space structure of $\textbf{B} \textbf{B}\mathbb{Z}$ is necessary.

Answer 1

I believe that working in the homotopy category and with $H$-space structures is not enough to define the correct category of algebras for the double dual monad. But in the language of $\infty$-categories, you are asking something like whether the adjunction $$ \operatorname{Map}(-,BB\mathbb{Z})\ \colon\ \mathrm{Grp}_{\mathbb{E}_1} \leftrightarrow \mathrm{Grp}_{\mathbb{E}_1}^{\mathrm{op}}\ \colon\ \operatorname{Map}(-,BB\mathbb{Z}) $$ is monadic. The $\infty$-categorical analogue of the Barr-Beck monadicity theorem [Higher Algebra Thm 4.7.3.5] gives you criteria to check this. It amounts to checking that the left adjoint preserves certain colimits (this seems ok here), but also that it is conservative. This fails here (and I believe it makes any version of your question fail, even if you feel like the $\infty$-categorical formulation here is not faithful to your question): For example, $\operatorname{Map}_{\mathrm{Grp}_{\mathbb{E}_1}}(\Omega S^n, BB\mathbb{Z}) \simeq \operatorname{Map}_{\mathcal{S}_*}(S^n, B^3\mathbb{Z}) = 0$ for $n>3$. So there are a lot of $\mathbb{E}_1$-groups (or $H$-spaces) whose maps into $BB\mathbb{Z}$ are trivial.