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Watch a 56-minute Topos Institute Colloquium talk exploring the mathematical concepts of Cauchy completeness and adjoints in double categories. Delve into Lawvere's 1973 paper observations about metric spaces as categories enriched in extended reals and their Cauchy completeness in relation to bimodule right adjoints. Learn about Paré's 2021 work on double categories and the connection between right adjoints of (S,R)-bimodules and finitely generated projective S-modules in commutative rings. Examine examples of Cauchy complete objects in various double categories and discover how projectivity results integrate with earlier theorems, extending applications to non-commutative rings, rigs, and quantales.
