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Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it. In the paper Joel write …

There's Russell's example from 1919, see here where conjugacy between relations is expressed diagrammatically.

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 …

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 …

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 …

I. Bakovic, Grothendieck construction for bicategories.

Since the time when Denis referred in the comments to the relevant nLab page, there has been a new proposal written up by Mike Shulman there: An elementary $(\infty,1)$-topos is an $(\infty,1)$-categ …

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 …

Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order …

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 …

Final coalgebras for functors on measurable spaces, Lawrence S. Moss and Ignacio D. Viglizzo: "We prove that every functor on the category Meas of measurable spaces built from the identity and consta …

Try Mike Shulman's page.

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

We have a case of relative cohesion used in an algebraic geometric setting discussed at the nLab. The entry for differential algebraic K-theory interprets Ulrich Bunke, Georg Tamme, Regulators and …

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