If A is some E_1 ring, is it true that both HH_.(A) and HH^.(A) are E2 modules over HH^.(A) but not A itself? (HH^. is hochschild cohomology and HH_. is hochschild homology). I know HH_. is a E2 module for sure, I'm guessing HH^. is too because any E2 ring is E2 module over itself, and I'm not sure about A carrying a E2 HH^. -module structure or not

@DenisNardin @AaronMazel-Gee I'm not sure if the TR-trace paper covers this too, but there are twisted Frobenii for THH with coefficients in a bimodule; this is worked out in a recent preprint by Tyler Lawson. Something interesting is that these twisted Frobenii actually make up the graded cyclotomic structure on THH(A+M).

@davik I think one issue here is that if A is only E_1 then we cannot talk about E_2 modules over it

@dhy @AlexanderCampbell There's a functor $\Delta^n\to \Theta_n$, so restriction along it will produce a presheaf of abelian groups on $\Delta^n$, which are $n$-fold chain complexes of abelian groups.

@dhy there is some attempt to describe abelian presheaves on $\Theta_n$ by C. Berger here: math.unice.fr/~cberger/Dold-Kan.pdf