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 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 ga 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. (C) 2000 Academic Press. [References: 19]
展开▼
机译：形式单子论的理论可以在任何2类中发展，但是当涉及到伪单子时，则被迫从2类转变为Gray类（半严格的3类）。 F. Marmolejo已采取了开发假单核形式理论的第一步，在这里我们继续该程序。我们展示了一个灰色类别的Psm，这样，从Psm到灰色类别A的灰色函数正好是A中的伪单子。这可以看作是伪单原子的完整相干结果。然后，我们根据灰色丰富分类理论意义上的加权极限，来描述拟单子的拟代数，拟代数的态射等。我们还展示了一个灰色类别的Psa，使得从Psa到ga的灰色函数正好是A中的伪附加物，表明每个伪附加物都诱导出一个伪monad，并且只要A承认所引用的限制，每个伪monad都会由伪附加物引起。到上面。最后，我们在a中定义了伪monads的灰色类别PSM（A），并表明它包含A作为完全反射子类别，当且仅当A接受这些相同限制时，它才是核心反射性的。 （C）2000年学术出版社。 [参考：19]
展开▼
