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]