I'm starting to read Higher Algebra to learn how to define an $E_1=A_{\infty}$ algebra in a monoidal $\infty$-category and how to define a $E_{\infty}$ algebra in a symmetric monoidal $\infty$-category. As far as I understand through reading around, the $A_{\infty}$ algebra can be defined without operads relatively tamely? But to define commutative algebras this is harder? Is this interpretation correct? What is the simplest way (least machinery way?) to define $A_\infty$ or $E_\infty$ algebras? And what is the reason why $E_\infty$ algebras require the operadic machinery? Apologies for this vague question which can probably be found in Higher Algebra.
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2$\begingroup$ A symmetric monoidal infty category is a cocartesian fibration over Fin_*. An E-infty algebra in there is a section which sends ‘collapse maps’ associated to inclusions of finite sets (ie inert maps) to cocartesian lifts thereof. $\endgroup$– Dylan WilsonDec 13, 2019 at 19:27
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3$\begingroup$ Alternatively: a symmetric monoidal infty category is a product preserving presheaf of infty-categories on Span(Fin), and symmetric monoidal functors are natural transformations. You can explicitly build a product-preserving presheaf associated to finite sets under disjoint union. An E_infty algebra is then a symmetric monoidal functor from Fin. $\endgroup$– Dylan WilsonDec 13, 2019 at 19:29
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2$\begingroup$ The issue, as @DylanWilson alludes, is that to define an $E_1$-or $E_\infty$-algebra in an $\infty$-category $C$, you first need a monoidal (resp symmetric monoidal) structure on $C$, and describing this requires at least as much machinery as describing the algebra. $\endgroup$– Charles RezkDec 14, 2019 at 15:34
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1$\begingroup$ In the $\infty$-setting, there is no meaningful distinction between A-infinity algebras and associative monoids. $\endgroup$– Charles RezkDec 14, 2019 at 15:38
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3$\begingroup$ @davik you don't necessarily need the machinery of operads in total, but you need some machinery. The issue is that it does not make sense to ask whether a category or a groupoid or something "is" symmetric monoidal (even classically) since one must supply a bunch of structure before one can ask that question. Classically, you can supply a notion of tensor product and a few natural transformations, and then you can ask "is this thing a symmetric monoidal category?". In $\infty$-land, it's structure all the way up, though there are neat ways to package this (a la Segal) that don't $\endgroup$– Dylan WilsonDec 14, 2019 at 16:54
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