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Functors of sub-descent type and dominion theory

Author: P. B. Johnson
Journal: Proc. Amer. Math. Soc. 122 (1994), 387-394
MSC: Primary 18A40
MathSciNet review: 1201297
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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.

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