nLab monadic cohomology

Idea

Every monad induces an augmented cosimplicial endofunctor; in an additive category we can produce a cochain complex out of any such cosimplicial endofunctor when applied to any particular object. Taking the cohomology of that complex yields monadic cohomology, see here at canonical resolution.

Related concepts

• cohomology

• canonical resolution

References

On monadic cohomology and comonadic homology:

• Jonathan M. Beck: Triples, algebras and cohomology, PhD thesis, Columbia University (1967), reprinted as: Reprints in Theory and Applications of Categories 2 (2003) 1-59 [tac:tr2, pdf]

• Michael Barr, Jon Beck, Homology and Standard Constructions, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Maths. 80, Springer (1969), Reprints in Theory and Applications of Categories 18 (2008) 186-248 [TAC:18, pdf]

• John Duskin, Simplicial methods and the interpretation of “triple” cohomology“ Memoirs of the AMS 163, Amer. Math. Soc. (1975) [ISBN:978-1-4704-0645-5]

For algebras, in relation to Hochschild cohomology:

• Michael Barr, Cohomology in tensored categories, Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 344–354 [doi:10.1007/978-3-642-99902-4_17, pdf, pdf]

• Michael Barr, Shukla cohomology and triples, J. Algebra 5 2 (1967) 222–231 [doi:10.1016/0021-8693(67)90036-1, pdf, pdf]

• Michael Barr, Harrison homology, Hochschild homology and triples J. Algebra 8 3 (1968) 314–323 [doi:10.1016/0021-8693(68)90062-8, pdf, pdf]

• Michael Barr, Cohomology and obstructions: commutative algebras, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Maths. 80, Springer (1969), Reprints in Theory and Applications of Categories 18 (2008) 357–374 [TAC:18, pdf, pdf]

See also:

• Michael Barr, Cartan-Eilenberg cohomology and triples, J. Pure Applied Algebra 112 3 (1996) 219–238 [doi:10.1016/0022-4049(95)00138-7, pdf, pdf]

• Michael Barr, Algebraic cohomology: the early days, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43 (2004) 1–26 [doi:10.1090/fic/043, pdf, pdf]