A Coherent Approach to Pseudomonads

✍ Scribed by Stephen Lack

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 350 KB
• Volume: 152
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1881

✦ Synopsis

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 Graycategories (semistrict 3-categories). The first steps in developing a formal theory of pseudomonads have been taken by F. 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, in terms of a weighted limit in the sense of Gray-enriched category theory. We also exhibit a Gray-category Psa such that a Gray-functor from Psa 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 referred to above. Finally we define a Gray-category PSM(A) of pseudomonads in A and show that it contains A as a full reflective subcategory, which is coreflective if and only if A admits these same limits.