Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $O$ be an operad (for example, $O$ could be an $A_m$ or $E_n$ operad or a tensor product thereof, and $\mathcal C$ could be spaces or spectra). There is a natural comparison functor $ho(Alg_O(\mathcal C)) \to Alg_O(ho(\mathcal C))$, where $ho(-)$ denotes taking the homotopy category; this comparison functor is generally very far from being an equivalence.
Moreover, if $\mathcal C$ is presentable, then generally $ho(\mathcal C)$ is not presentable, and likewise $Alg_O(ho (\mathcal C))$ is not presentable. One might ask, though, for some "refinement" $\mathcal D$ of $Alg_O(ho (\mathcal C))$ which is presentable:
Question 1: If $\mathcal C$ is presentable, does there exist a presentable (or at least accessible) $\infty$-category $\mathcal D$ such that $ho(\mathcal D) = Alg_O(ho (\mathcal C))$?
Motivation: Consider the homotopy commutative ring spectrum $MU$. Then $MU$ has a universal property in the category $Alg_{Comm}(ho(Spectra))$. But because $Alg_{Comm}(ho(Spectra))$ is not presentable, there's no "general construction of universal objects" from which this universal property follows -- it must be verified to hold "by hand". By contrast, $MU$ has a natural universal property as an $E_\infty$ ring spectrum which follows directly from its construction as a Thom spectrum, as explained by Hopkins and Lawson (but it's not as simple as being the universal complex oriented thing). And on the other hand, it follows by abstract nonsense that a universal complex-oriented $E_\infty$ ring spectrum $MX_1$ exists (but $MX_1 \neq MU$). It would be nice if the existence of a universal complex-oriented homotopy commutative ring spectrum were as formal as the existence of the $E_\infty$ ring spectrum $MU$ or the $E_\infty$ ring spectrum $MX_1$. It would then be a computation to verify that the universal complex oriented homotopy commutative ring spectrum is actually $MU$.
Example: The simplest nontrivial example I can think of is when $O = E_0$ and $\mathcal C = Spaces$. Then $Alg_{E_0}(Spaces) = Spaces_\ast$ is the $\infty$-category of pointed spaces. But $Alg_{E_0}(ho(Spaces))$ is the category of spaces equipped with a basepoint in the homotopy category. This is equivalent to looking at objects in the homotopy category of spaces equipped with a distinguished connected component, and homotopy classes of maps preserving such. In this case, there is a natural accessible $\infty$-category $\mathcal D$ refining $Alg_{E_0}(ho(Spaces))$. Namely, take $\mathcal D = Sets_\ast \times_{Sets} Spaces$ (where the functor $Spaces \to Sets$ is $\pi_0$). Since all the categories and functors involved are acessible, $\mathcal D$ is also accessible. But I don't believe it's presentable. And this trick doesn't immediately generalize to other cases -- it relies on having an alternate description of what an $E_0$ structure in the homotopy category is (it doesn't even generalize to the case $O= E_0$, $\mathcal C = Spectra$).
This trick actually suggests to me that "spectra equipped with a multiplication which is associative / commutative on homotopy groups" might be a more natural category to think about than actual homotopy associative / commutative ring spectra, because a trick like this will work to give an accessible refinement of the category! A relevant case of this might be Morava $K$-theory $K(n)$ at the prime 2, whose multiplication is not homotopy commutative, but is commutative on homotopy groups.
The above framework encompasses categories like homotopy associative or homotopy commutative ring spectra, but I think does not quite encompass $H_\infty$ ring spectra. But $H_\infty$ structures are still very operadic in nature: given an operad $O$, define an $HO$-algebra in $\mathcal C$ to be an object $X \in \mathcal C$ equipped with maps $O(n) \otimes_{\Sigma_n} X^{\otimes n} \to X$ such that the appropriate diagrams commute in $ho(\mathcal C)$. Then an $H_\infty$ structure is the same thing as an $HE_\infty$ structure. Morphisms are morphisms in $ho(\mathcal C)$ making the appropriate diagrams commute in $ho(\mathcal C)$.
Question 2: If $\mathcal C$ is presentable, is there a presentable (or at least accessible) $\infty$-category $\mathcal E$ such that $ho(\mathcal E) = Alg_{HO}(\mathcal C)$?
