# Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose totalization computes the (global sections of the structure sheaf of the...) quotient of $Spec(X)$ under the action of $Spec(H)$, if $X$ is a commutative ring spectrum. In degree $n$ it looks like $X\wedge H\wedge\ldots H$, with $n$ copies of $H$. The coface maps are given by either the coaction of $H$ on $X$, the diagonal map of $H$, or the unit map of $H$. Codegeneracies are given by either applying the counit of $H$ or the multiplication of $H$. This can be thought of as an explicit construction for the "cotensor product" of $X$ and $\mathbb{S}$ as $H$-comodules. This can in fact be made quite explicit if one works in a suitable framework for coalgebraic structure (e.g. this).

On the other hand, if $X$ is a commutative ring spectrum, there is something called the Amitsur complex for $X$. In degree $n$ this has the form of $X\wedge X\wedge\ldots\wedge X$, $n+1$-copies of $X$. The coface maps here are given by using the unit of $X$, and the codegeneracies are given by the multiplication on $X$. This is not really given as any kind of tensor product, but rather as the effect of iterating a monad (the one given by tensoring with $X$) on $\mathbb{S}$ over and over again and considering the resulting cosimplicial resolution.

In Proposition 12.1.8 of the aforementioned monograph (available here), Rognes proves that if we have an equivalence $X\wedge X\to X\wedge H$ given by $$X\wedge X\overset{1\wedge\Delta}\to X\wedge X\wedge H\overset{\mu\wedge 1}\to X\wedge H,$$ we can iterate this at each level to produce a cosimplicial equivalence between the two cosimplicial objects I described above (see Definition 12.1.7 op. cit.). He remarks that a diagram chase is necessary to show that this map is in fact cosimplicial, and moreover that it uses strict coassociativity and counitality of $H$.

My question is the following: can one deduce the above equivalence in the setting of a quasicategories when one does not have recourse to a diagram chase, especially not one requiring strict commutativity of diagrams? It would seem that doing this in this setting would require having a deeper understanding of what those two cosimplicial objects are.

• Don't have anything useful, but: (1) this looks like a shearing map showing something or other is a torsor. I vaguely remember similar things appearing in the Galois descent part of DAG XI. Of course Luries notion of étale is more restrictive but maybe this diagram appears. (2) when dealing with cosimplicial objects in infty categories you can build maps in a pretty straightforward way that's not too far from classical category theory-see the appendix of HTT when he talks about reedy categories Jul 19, 2015 at 1:27
• Au contraire @DylanWilson I think both of those comments might be immensely useful. Thanks! Jul 19, 2015 at 1:57
• Oh, and it is precisely a shearing map, I'm just trying to build it "coherently" in some sense. Jul 19, 2015 at 1:58

The basic idea is to recognize that the comonad underlying the Amitsur complex is the initial comonad coacting on $$X$$. As a result, there is a comonad map from it to the comonad associated to the bialgebra $$H$$. This induces the relevant cosimplicial map which one then checks is an equivalence in the case of Thom spectra.