Functors of Subdescent Type and Dominion Theory

Necessary and sufficient conditions are given for the EilenbergMoore comparison functor <1> 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) : C°p -» 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 domfi 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 «¡generator for the category doing , in which case dom^ is the reflective hull of B in C and, moreover, the category doms 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 .