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The 2-category of categories, adjunctions, and conjugate natural transformations (i.e., maps of adjunctions between the same categories) is used in an approach to modal type theory in Adjoint Logic wi …

One further line of response would again invoke Rota: "What can you prove with exterior algebra that you cannot prove without it?" Whenever you hear this question raised about some new piece …

Did you follow the thread Simon Willerton started on profunctors between metric spaces, which took us all the way to optimal transport theory?

I'm interested in situations where universal objects come with more structure than their definitions suggest. A classic case of this is where the free abelian group on one element has a ring structure …

I. Bakovic, Grothendieck construction for bicategories.

We had a chat about this topic over here, prompted by remarks by David Kazhdan.

Mike Shulman has been thinking about n-toposes in general and 2-toposes in particular.

Perhaps you know that over the years we've had many discussions about matrix mechanics at the Cafe. Depending on the rig (ring without negatives) used, you end up with a different form of mechanics - …

Theorem 7.2 of Kirk Sturtz, The factorization of the Giry monad and convex spaces as an extension of the Kleisi category, provides one answer for measurable spaces.

There is a setting as outlined here, where by composing left and right adjoints to the pullback with the pullback, one can form an adjunction between a comonadic necessity and a monadic possibility op …

There's also the work of Francis Borceux and Marco Grandis, Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra 208 (2007), no. 2, 665-689, doi. There t …

For a recent approach that looks to provide a better categorical environment for probability theory: Chris Heunen, Ohad Kammar, Sam Staton, Hongseok Yang, A Convenient Category for Higher-Order Prob …

Belatedly, an answer in set-based situations to What would be the corresponding slogan to "no junk, no confusion" for final coalgebras? Given an initial algebra, any algebra will have a special su …

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory f …

There's an interesting variant of your question, which may perhaps have been included in the first part of it, as to whether there are parts of mathematics where categories have little traction, and w …