We say a set-valued functor on a category is nearly representable if it is a quotient of a representable functor by a group of automorphisms. A distributor is a set-valued functor in two arguments, contravariant in one argument and covariant in the other. We say a distributor is slicewise nearly representable if it is nearly representable in either of the arguments whenever the other argument is fixed. We consider such a distributor a weak analogue of adjunction. Under a finiteness assumption on the domain categories, we show that every slicewise nearly representable functor is a composite of two distributors, each of which may be considered as a weak analogue of (co-)reflective adjunction.
Keywords: distributor, adjoint, nearly representable
2010 MSC: 18A40
Theory and Applications of Categories, Vol. 34, 2019, No. 31, pp 993-1035.