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I'm not very familiar with dg algebras (not necessarily commutative) and I'm wondering if any $E_1$ algebra in the sense of infinity categories (i.e. monoid in the stable category of $R-Mod$) over $R$ some commutative ring (discrete) can be described by a dg algebra over R? Specifically I have some $E_1$ algebra in mind which has homotopy groups $\mathbb{F}_p$ in homological degrees $0$ and $-1$. I'm wondering if the $E_1$ structure has to be the "obvious" one. I think there may be other $E_1$ structures on it but I'm not sure how to see this.

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    $\begingroup$ Dold Kan is not enough right? For nonconnective spectra $\endgroup$
    – davik
    Mar 15 at 3:29
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Yes, this is precisely the content of Theorem 7.11 in arXiv:1410.5675, which should be combined with §7.4 of arXiv:1510.04969. In fact, the cited results prove this for any nonsymmetric operad in chain complexes over a commutative ring, and are also applicable to ∞-categories other than chain complexes.

In the case of symmetric operads (such as ${\rm E}_∞$) the result continues to hold for characteristic 0, i.e., rational chain complexes. For symmetric operads in characteristic $p$ there are algebras that cannot be rectified as stated, and one must use other categories instead of chain complexes, e.g., simplicial modules over ${\bf F}_p$ (see §7.3 of arXiv:1510.04969).

In all such results, there are two main ingredients. The first ingredient shows that any ∞-algebra over an ∞-operad $O$ can be rectified to a strict algebra over a strict operad $Q$, where the underlying ∞-operad of $Q$ is equivalent to $O$. This roughly amounts to the free strict $Q$-algebra computing the free ∞-algebra over $O$, which in its turn almost immediately boils down to the strict coinvariants of tensor products $Q_n⊗_{Σ_n}X^{⊗n}$ computing the corresponding homotopy coinvariants of derived tensor products, for any cofibrant object $X$. The main property required here is that $Q_n$ should be cofibrant (projectively cofibrant with respect to the $Σ_n$-action in the case of symmetric operads).

The second ingredient shows that any strict $Q$-algebra can be rectified to a strict $R$-algebra, where $R$ is the operad we are interested in, e.g., the associative or commutative operad, and $f\colon Q→R$ is a weak equivalence of operads. The main property required here is that $f_n ⊗_{Σ_n} X^{⊗n}$ should be a weak equivalence for any cofibrant object $X$. This can happen for two reasons: either the maps $f_n$ are sufficiently nice (e.g., their source and target are $Σ_n$-projectively cofibrant) or the category in which we work is nice. The latter holds, for example, for rational chain complexes and for various categories of symmetric spectra.

For nonsymmetric operads, such as the associative operad, we can drop $Σ_n$, which makes the condition much easier to satisfy in practice. In particular, it is always true for simplicial sets, chain complexes over any commutative ring, topological spaces, simplicial modules, simplicial presheaves, etc. In all these cases, ∞-algebras over nonsymmetric operads can always be rectified to strict algebras.

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    $\begingroup$ Intuitively, I know that in case the operad is free (or non-symmetric), one can forget about the more elaborate simplicial or $\infty$-theoretical approaches to the homotopy theory of its algebras and just look at dg objects, but I do not know of a sketch of proof (though I know of counterexamples in positive char). Could you offer such a sketch here? Is it simply because the associated Schur functor is exact, and this is enough? $\endgroup$ Mar 15 at 7:45
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    $\begingroup$ @davik: Yes. For example, E_1-rings can be rectified to (strict) monoids in simplicial symmetric spectra. Thus, the slice category of F_p rectifies to the slice category of H(F_p), the Eilenberg–MacLane symmetric simplicial spectrum of F_p. The latter slice category is equivalent to the category of monoids in modules over H(F_p). The category of modules over H(F_p) is Quillen equivalent to chain complexes over R. Now we invoke Theorem 8.10 in arXiv:1410.5675 to show that monoids in H(F_p)-modules are Quillen equivalent to algebras over an operad P. $\endgroup$ Mar 15 at 19:13
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    $\begingroup$ @DmitriPavlov I believe that the slice category in question is equivalent instead to the category of monoids in bimodules over H(F_p). $\endgroup$ Mar 16 at 3:13
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    $\begingroup$ @TylerLawson: I think I misread "not central" as "central" when I wrote the above comments. Of course, if we do not assume F_p to be central, then one gets bimodules instead of modules. $\endgroup$ Mar 16 at 4:28
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    $\begingroup$ @davik: My comments above apply to the central case, since I thought you were asking a question about an equivalent category. If you really are interested in noncentral F_p-algebras instead, these can be rectified to monoids in bimodules over HF_p in symmetric simplicial spectra (say). But I don't know a good chain-level model for HF_p-bimodules, since the other action can act by Steenrod powers, I think. $\endgroup$ Mar 16 at 4:41

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