Hmmmm, can one generalize directly from this definition to comodules over a comonoid? For instance, for objects in C with a G-action, we can look at functors BG->C, but what is the “classifying object” for comodules? In general, one would like to set up the entire apparatus of colored operad type objects identifying comonoids and their comodules from the standpoint of a fibration of categories, I think. Does that all go through if one just puts “op” at the end of everything?

]]>Comonoids are just monoids in the opposite category. Why doesn’t that work for you?

]]>Hey all,

So I’ve been kind of bugging out trying to find some kind of coherent theory of comonoids in $\infty$-categories. This, for instance, would apply to comonads (as co-associative comonoids in endomorphism categories) among other things. When I try to use Lurie’s stuff, I end up having to trace further and further back to try to prove anything, and end up feeling like I need a theory of cooperads. Somehow comonoid structures seem fundamentally different than monoid structures. Does anyone know how to do this, or if it’s written down clearly anywhere? For instance, Lurie has this nice theorem in one of the DAGs where he shows that monoids are essentially simplicial objects, and this seems to generalize pieces of Emily Riehl’s work with Dominic Verity, except for the fact that there’s no analog for comonoidal objects. It’d be nice to have the analogous statement saying that comonoids are cosimplicial objects in some essential way.

Thanks for any ideas!

-Jon

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