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A general coherence result. (English) Zbl 0668.18010

Commencing with a paper by S. MacLane in 1963, results about certain diagrams commuting were seen as constituting the essence of a coherence theorem. In 1972/73, G. M. Kelly questioned the importance of such results and outlined several new approaches to coherence. Following Kelly’s footsteps, the author states and proves a general coherence result that subsumes earlier results in the literature. The theorem asserts that, for a large class of 2-monads, every pseudo-algebra is equivalent, in the appropriate 2-category of algebras, to a strict algebra.
Reviewer: J.Sonner

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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References:

[1] R. Blackwell, G.M. Kelly, and A.J. Power, Two-dimensional monad theory, submitted.; R. Blackwell, G.M. Kelly, and A.J. Power, Two-dimensional monad theory, submitted. · Zbl 0675.18006
[2] Kelly, G. M., An abstract approach to coherence, (Coherence in Categories. Coherence in Categories, Lecture Notes in Mathematics, 281 (1972), Springer: Springer Berlin), 106-147 · Zbl 0243.18016
[3] Kelly, G. M., On clubs and doctrines, (Category Seminar Sydney 1972/73. Category Seminar Sydney 1972/73, Lecture Notes in Mathematics, 420 (1974), Springer: Springer Berlin), 181-296 · Zbl 0334.18018
[4] Kelly, G. M., Doctrinal adjunction, (Category Seminar Sydney 1972/73. Category Seminar Sydney 1972/73, Lecture Notes in Mathematics, 420 (1974), Springer: Springer Berlin), 257-280 · Zbl 0334.18004
[5] Kelly, G. M., Coherence theorems for lax algebras and for distributive laws, (Category Seminar Sydney 1972/73. Category Seminar Sydney 1972/73, Lecture Notes in Mathematics, 420 (1974), Springer: Springer Berlin), 281-375 · Zbl 0334.18014
[6] Kelly, G. M., Some remarks on categories with structure, (Sydney Category Seminar Reports (1978), University of Sydney) · Zbl 0436.18006
[7] Kelly, G. M.; Street, Ross, Review of the elements of 2-categories, (Category Seminar Sydney 1972/73. Category Seminar Sydney 1972/73, Lecture Notes in Mathematics, 420 (1974), Springer: Springer Berlin), 75-103 · Zbl 0334.18016
[8] MacLane, S., Natural associativity and commutativity, Rice University Studies, 49, 28-46 (1963) · Zbl 0244.18008
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