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Callan McGill (Nov 07 2022 at 15:24):Does anyone know of something that describes some of the theory of indexed adjunctions? I have in mind something like the situation of Fam(Set) and Fam(Vect) where there is a fiberwise indexed adjunction between the two.
Jade Master (Nov 07 2022 at 16:16):Given a category C, there is a 2-category [C,Cat] of functors from C into Cat, natural transformations between functors, and modifications between natural transformations. An adjunction internal to this 2-category is the same thing as an indexed family of adjunctions between the fibers.
Jade Master (Nov 07 2022 at 16:26):And then also there's an equivalence via the Grothendieck construction. The equivalence will preserve adjunctions, so it's saying that you can glue the fibrewise adjunctions into a single adjunction on their Grothendieck constructions.
fosco (Nov 07 2022 at 17:26):Do you have in mind something more specific / extended than 1.1.8 in Jacobs' book on Categorical Logic? (The section in question is called "fibrewise structure and fibred adjunctions". The section also points out a subtlety one has to consider: one can glue a family of adjunctions between the fibers if a Beck-Chevalley condition holds. image.png )
Jonathan Weinberger (Nov 07 2022 at 18:31):See also Borceux, Handbook of Categorical Algebra Vol. 2, Section 8.4 "Fibred adjunctions"
fibadj01.png
fibadj02.png
(Diagram 8.8 is the triangles for the triangle identities)
