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TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be?

For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). A comonad is a special comonoid, so by a universal property of $\Delta^{op}$ we get a cosimplicial object (the Bar construction [1]).

From this, one derives a cohomology theory of this algebraic theory. This subsumes group cohomology, Lie algebra cohomology, Hochschild cohomology, and Harrison's cohomology for commutative algebras [2, chapter 6+7].

Question

  1. What cohomology theories are known to not from comonads?
  2. Thinking of a group as a category with one object, this line of thoughts fits naturally into. Does it has an analogy to higher categories too?

Reference

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    $\begingroup$ How about singular cohomology? $\endgroup$
    – Qiaochu Yuan
    Commented Nov 19, 2020 at 18:09
  • $\begingroup$ I don't know much really.. isn't it a comonad cohomology in some way? $\endgroup$
    – Student
    Commented Nov 19, 2020 at 18:10
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    $\begingroup$ Godement resolution for computing sheaf cohomology can be viewed as coming from a particular adjunction, hence from a comonad. To give an example of some cohomology not coming from a comonad you have to come up with meaningful right derived functors of a functor which does not have an adjoint. There must be such examples except I cannot think of any right now $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 19, 2020 at 19:54
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    $\begingroup$ It seems by the adjunction functor theorem, many meaningful functors are adjoints naturally. Leave alone cohomologies! This is non-rigorous I know. I mentioned it just to strengthen my doubt. $\endgroup$
    – Student
    Commented Nov 19, 2020 at 20:06
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    $\begingroup$ Well, how about K-theory then? $\endgroup$
    – Qiaochu Yuan
    Commented Nov 19, 2020 at 20:38

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