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Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on a category $\mathcal{B}$ and a $\bot$-comodule $b$ with structure map $\theta$, he says that the 0th cohomology of $(b,\theta)$ is the group of comodule automorphisms $(b,\theta)\to (b,\theta)$. The 1st descent cohomology is the pointed set of isomorphism classes of $\bot$-comodule structures on $b$, if I'm reading him correctly. Mesablishvili goes on to show that if the comonad is the free-forgetful one on the category of $\top$-algebras for some monad $\top$, one can compute the descent cohomology of that comonad in terms of a certain Amitsur cohomology associated to the monad $\top$.

Many parts of this generalize quite nicely to the case of $\infty$-categories. Does anyone know if someone (I'm thinking Jacob Lurie primarily) has defined a similar sort of descent cohomology in the case of comonads on $\infty$-categories? I feel like this sort of thing could be somewhere in DAG or Higher Algebra. I believe that the analog is most certainly true in the homotopical setting, and that this is probably known to many experts. I only wonder if it's written down anywhere. If not, I guess I'll write it down.

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  • $\begingroup$ The short answer, for the record, is no. Nobody has done this. $\endgroup$
    – Jonathan Beardsley
    Commented Feb 3, 2014 at 4:44

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This seems highly related to the research program of Kathryn Hess. In 2010 she wrote a paper which lays the groundwork for homotopic descent and codescent, and that's where I would start if I were you. Much of this is in the language of model categories, though she also gives the $\infty$-category interpretation. In this paper she is interested in the cohomology descent spectral sequence. Is that one of the things you were looking for a higher analog of?

There was a later paper of Hess and Shipley in which very general conditions are given so that a model structure transfers to the category of coalgebras over a comonad. Hess has several other papers studying chain coalgebras and Hopf-Galois extensions, and there is a chance they might contain some of the general theory you are looking for. Good luck!

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  • $\begingroup$ Thanks David. I'm pretty familiar with that paper. Unfortunately, she does not construct a descent cohomology sort of thing in her work, though one could certainly see a way to do it in her framework. I'm hoping to find some way to do this in terms of $\infty$-categories just because I don't really know how to put model category structures on things that don't support obvious left Quillen functors from something I already know about. However, I think the idea is the same from both points of view. $\endgroup$
    – Jonathan Beardsley
    Commented Dec 20, 2013 at 18:42
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    $\begingroup$ I should add that Galois cohomology and Hopf-Galois cohomology descent spectral sequences (e.g. the Adams Novikov Spectral Sequence) both follow as a special case of the situation I'm describing. It's just that in those cases, the relevant equivalences allow one to identify the Amitsur cohomology as a group or co-group cohomology. $\endgroup$
    – Jonathan Beardsley
    Commented Dec 20, 2013 at 18:44
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    $\begingroup$ Where by group and co-group cohomology I really mean homotopy fixed points spectral sequence or homotopy co-fixed points spectral sequence (which is NOT homotopy orbits!!). $\endgroup$
    – Jonathan Beardsley
    Commented Dec 20, 2013 at 18:46
  • $\begingroup$ I don't know anything about Mesablishvili's work, and I guess I missed that this was the thing you wanted an $\infty$-categorical analogue of. Sorry my answer was something you already knew. I have no idea where in Lurie's work to search for this. Whenever I'm in that position I just ask Dylan or Akhil. $\endgroup$
    – David White
    Commented Dec 20, 2013 at 18:49
  • $\begingroup$ No worries, thanks for your answer! I think very few people are familiar with the notion of descent cohomology that I want to discuss. It's unfortunate, because I think it's an interesting notion. The story most certainly does not end with classifying morphisms which are effective descent morphisms. $\endgroup$
    – Jonathan Beardsley
    Commented Dec 20, 2013 at 21:17

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