Functors of sub-descent type and dominion theory

Abstract:

Necessary and sufficient conditions are given for the Eilenberg-Moore comparison functor $\Phi$ 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 $U = {\mathbf {C}}( - ,B):{{\mathbf {C}}^{{\text {op}}}} \to {\mathbf {Set}}$, 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 ${\mathbf {dom}_B} \subset {\mathbf {C}}$ of so-called B-dominions, consisting of certain canonically arising regular subobjects of powers of the object B. The functor $U = {\mathbf {C}}( - ,B)$ is of subdescent type if and only if the object B is a regular cogenerator for the category ${\mathbf {dom}_B}$, in which case ${\mathbf {dom}_B}$ is the reflective hull of B in C and, moreover, the category ${\mathbf {dom}_B}$ admits a Stone-like representation as (being contravariantly equivalent, via the comparison functor $\Phi$, to) a full, reflective subcategory of the category of algebras for the triple in Set induced by the functor U.