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

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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

+ [1] https://math.ucr.edu/home/baez/qg-spring2007/qg-spring2007.html#computation
+ [2] Acyclic Models-[Michael Barr]