A characterization of the image of Comod

Paddy McCrudden – 26 November 1997

Let V = k-Vect be the monoidal category of Vector Spaces. Recall that a one object V^op enriched category is precisely a k-coalgbra. Thus we consider the 2-category V^op-Cat to be a generalization of the category of k-Coalgbras. Let V-Act denote the 2-Category of V-Actegories. This is simply the 2-category T-psAlg for the pseudomonad T= V x - : Cat --> Cat. Thus an object of V-Act is a category A equipped with an action V x A --> A, which is associative and unital up to coherent isomorhism. Let V-Act//V denote the 2-Category whose objects are triples, (A, U : X ---> V-Act(A,V)), where A is a V-Actegory, X is a set and U is a function. Thus we may consider V-Act//V to have objects A in V-Act equipped with a family a "underlying" functors into V. We define a 2-functor Comod : (V^op-Cat)^op ---> V-Act//V which agrees with the usual definition if Comod when restricted to the one object V-Actegories (= coalgebras). That is, if C is acoalgebra, then Comod(C) is the usual category of representations together with its underlynig functor into V. We prove that Comod is 2-fully faithful, and that there exists a partially defined right 2-adjoint. thus the question arises: "which objects in V-Act//V are equivalent to objects of the form Comod(C), for some V^op-Category C." In this seminar, we prove that the objects in V-Act//V that are equivalent to objects of the form Comod(C) are precisely those objects for which the family of functors ((U_x:A--->V)_(x in X)) are comonadic in the sense of Street: 2-constructions on Lax functors. there are intimate connections of this theorem with Lawvere et als "structure and semantics." these connections have since been clarified.