Susan Niefield: "Cauchy Completeness and Adjoints in Double Categories"

Topos Institute Colloquium, 8th of February 2024.
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In his 1973 paper (TAC Reprints, 2002), Lawvere observed that a metric space Y is a category enriched in the extended reals, and showed that Y is Cauchy complete if and only if every bimodule (i.e., profunctor) with codomain Y has a right adjoint. More recently, Paré (2021) considered adjoints and Cauchy completeness in double categories, and showed that an (S,R)-bimodule M has a right adjoint in the double category of commutative rings if and only if it is finitely generated and projective as an S-module. It is well known that the latter property characterizes the existence of a left adjoint to tensoring with M on the category of S-modules, and this was generalized to rigs and quantales in a 2017 paper by Wood and the speaker.

This talk consists of two parts. First, after recalling the relevant definitions, we present examples of Cauchy complete objects in some "familiar" double categories. Second, we incorporate the two above mentioned projectivity results into a version of the 2017 theorem with Wood which we then apply to (not-necessarily commutative) rings, rigs, and quantales.