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Recent work by Qi Zhu, Fractured Structure on Condensed Anima, is looking to approach condensed mathematics via Jacob Lurie's concept of a fractured structure, seen as "local cohesive structure", cf. …

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 …

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 …

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 …

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 …

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 …

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.

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 …

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 …

You'll see more category theory, including adjunctions, relating to differential geometry by departing from standard topics and treatments, e.g., by looking to synthetic differential geometry or highe …

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

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 …

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 …

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 …

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 …