## Meditation on pseudomonads

### Steve Lack – 2 September 1998

These talks were based on the paper A coherent approach to pseudomonads.
Abstract: The formal theory of monads can be developed in any 2-category, but when it comes to pseudomonads, one is forced to move from 2-categories to Gray-categories (semistrict 3-categories). The first steps in developing a formal theory of pseudomonads have been taken by Fransisco Marmolejo (see his papers Doctrines whose structure forms a fully faithful adjoint string and Distributive laws for pseudomonads, both published in and available from Theory and Applications of Categories), and we continue that program.
We exhibit a Gray-category Psm such that a Gray-functor from Psm to a Gray-category A is precisely a pseudomonad in A; this may be viewed as a complete coherence result for pseudomonads. We then describe the pseudoalgebras for a pseudomonad, the morphisms of pseudoalgebras, and so on, in terms of a weighted limit in the sense of Gray-enriched category theory.
We also exhibit a Gray-category Psadj such that a Gray-functor from Psadj to A is precisely a pseudoadjunction in A, show that every pseudoadjunction induces a pseudomonad, and that every pseudomonad is induced by a pseudoadjunction, provided that A admits the limits mentioned in the previous paragraph.
Finally we define a Gray-category Psm(A) of pseudomonads in A, show that it contains A as a full reflective subcategory, which is coreflective if and only if A admits these same limits.

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