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John Baez (Apr 08 2020 at 22:44):Christian Williams gave a talk at UCR yesterday:
Abstract. This is an introduction to the role of adjoints in categorical logic. The logical operations "or" and "and" are left adjoints of the diagonal map on propositions. More generally, existential and universal quantifiers can be seen as left and right adjoints to substitution. In a similar way, composition with a function between sets defines a functor between their slice categories that has both left and right adjoints. This in turn is a special case of how in any locally cartesian closed category, any morphism induces a functor between slice categories that has both left and right adjoints.
https://www.youtube.com/watch?v=uQp-Pi5jSNk
Christian Williams (Apr 09 2020 at 02:37):I went pretty slow at the beginning - feel free to skip ahead. We get up to "locally cartesian closed iff pullback has a right adjoint", but we didn't have time for the proof. Maybe next time.