I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative Hopf-algebras in this setting as functors $H:Alg(\mathbb{S})\to E_\infty Spc$, in other words, functors from associative algebras that land in the $\infty$-category of infinite loop spaces. This, in my mind, generalizes the notion of what one does to produce a cocommutative cogroup object in algebra. One of the things I'd like to get from this definition is that if we have a one-old loop space $\Omega X$, then its suspension spectrum should be such a functor (with the cogroup structure coming from the diagonal on $\Omega X$ and the algebra structure coming from the fact that it's a loop space). However, from a bit of discussion in the homotopy theory chat room, it seems that this is not attainable by the above given definition.

Another idea suggested in the homotopy theory chat room was the notion of some kind of cosimplicial object in associative algebras. Embarassingly, this also sort of trips me up, since cosimplicial objects look more like monoids than comonoids to me (e.g. we take a space which is naturally a comonoid and apply the cobar construction to get a monoid, a.k.a. loops on that space).

Do people have workable notions for this idea, or definitions that they know are written down anywhere? One can do relatively straightforward things in $E_\infty$-rings because the tensor product is the coproduct, but this is not the case for associative ring spectra.

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So, okay there are clearly some things wrong with what I've said above, and some bad intuition (e.g. my confusion about the cosimplicial structure, and about the functorial point of view for associative algebras) but I would still very much like to know if anyone has a good working definition for Hopf-algebras in associative ring spectra and moreover whether or not this includes the sort of example I give above.

nota cocommutative cogroup object in algebras even in the ordinary setting, precisely because the tensor product of associative algebras is not the coproduct. What is true is that a commutative Hopf algebra is a cogroup object in commutative algebras, and that a cocommutative Hopf algebra is a group object in cocommutative coalgebras. $\endgroup$