Functors of sub-descent type and dominion theory

Necessary and sufficient conditions are given for the EilenbergMoore comparison functor 4) arising from a functor U (having a left adjoint) to be a Galois connection in the sense of J. R. Isbell, in which case the functor U is said to be of subdescent type. These conditions, when applied to a contravariant hom-functor U = C(-, B): COP Set, read like a kind of functional completeness axiom for the object B. In order to appreciate this result, it is useful to consider the full subcategory domB C C of so-called B-dominions, consisting of certain canonically arising regular subobjects of powers of the object B. The functor U = C(-, B) is of subdescent type if and only if the object B is a regular cogenerator for the category domB , in which case domB is the reflective hull of B in C and, moreover, the category domB admits a Stone-like representation as (being contravariantly equivalent, via the comparison functor 4), to) a full, reflective subcategory of the category of algebras for the triple in Set induced by the functor U.