From a model theoretic perspective, these are the classes from which we can build generalized indiscernibles and generalized Ehrenfuecht-Mostowski models in $\mathbb$, where the local objects are described via the right adjoint.
From a combinatorial set theory perspective, these are the classes that generalize the Erdos-Rado Theorem in the same way that Ramsey Classes generalize Ramsey's Theorem.
We present preliminary results on Erdos-Rado classes. What categories characterize large, finitely accessible categories by faithfully and colimit-preservingly embedding into them? I will give examples where the language of category theory allows for elegant (re)formulations of theorems of Hodges and Shelah, as well as examples where category theory cannot seem to go on its own.Īlmost all the material in this talk is based on my collaboration with Jiri Rosicky as well as conversations with Michael Makkai. I will give a (necessarily partial and subjective) overview of where the ideas of Makkai and Pare have taken us during these three decades, and where they might take us in the future. The 1989 monograph of Makkai and Pare, "Accessible categories: the foundations of categorical model theory" provides, I believe, both the most systematic and most beautiful framework for category theory to serve model theory, and vice versa. Starting with Lawvere's "functorial semantics" in the 1960's, category theory coexisted (and sometimes, competed) with Tarski-style model theory to serve as a language of metamathematics. Titles & abstracts Joan Bagaria Some considerations around the Weak Vopenka Principle Tibor Beke Accessible categories and model theory: hits and misses
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