A coherent approach to pseudomonads

Stephen Lack

This material has been published in Advances in Mathematics 152:179-202, 2000, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Academic Press. This material may not be copied or reposted without explicit permission.

It is also available via IDEAL (International Digital Electronic Access Library).

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 Francisco Marmolejo, and here 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, as 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 a pseudoadjunction provided that A admits the limits mentioned in the previous paragraph. Finally we define a Gray-category \psma 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.


Click here to download a gzipped postscript file of the complete paper.


Steve Lack
Last modified: Wed Sep 13 08:55:33 EST 2000