Accessible categories, set theory, and model theory: an invitation

Vasey, Sebastien

doi:10.48550/arxiv.1904.11307

Cited by 3 publications

(3 citation statements)

References 37 publications

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“…Synthesizing the works of Beke, Rosický, Lieberman, Vasey and others (cf., e.g., [6,[24][25][26][27]35]) in a sentence, accessible categories with directed colimits generalize Shelah's framework of abstract elementary classes, and are special enough to recover the main features of categorical model theory.…”

Section: Mathematical Logic Quarterlymentioning

confidence: 99%

Formal model theory and higher topology

Di Liberti

2024

Mathematical Logic Qtrly

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We study the 2‐categories BIon, of (generalized) bounded ionads, and , of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex) geometric sketches, which serve as categorical specifications of geometric theories. We provide reconstruction and completeness‐like results. We relate abstract elementary classes to locally decidable topoi. We introduce the notion of categories of saturated objects and relate it to atomic topoi.

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Section: Mathematical Logic Quarterlymentioning

confidence: 99%

Formal model theory and higher topology

Di Liberti

2024

Mathematical Logic Qtrly

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“…It was later shown by Lieberman [Lie09] and independently by Rosický and Beke [BR12] that abstract elementary classes are accessible too. The reader that is interested in this connection might find interesting [Vas19a], whose language is probably the closest to that of a model theorist.…”

Section: A3 Locally Presentable Categories and Essentially Algebraic ...mentioning

confidence: 99%

Formal Model Theory & Higher Topology

Di Liberti

2020

Preprint

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We study the 2-categories of (generalized) bounded ionads BIon and accessible categories with directed colimits Accω as an abstract framework to approach formal model theory. We relate them to topoi and (lex) geometric sketches, which serve as categorical specifications of geometric theories. We provide reconstruction and completeness-like results. We relate abstract elementary classes to locally decidable topoi, and categories of saturated objects to atomic topoi. Contents

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“…O O identified as "independent." That this specializes to stable independence in abstract elementary classes is proven in [16,Section 8]; connections with the classical notion are examined in, e.g., [27,Example 5.7(6)]. It is shown in [16], moreover, that this axiomatization is canonical in accessible categories with monomorphisms, assuming that they have chain bounds, thereby extending the canonicity theorem for abstract elementary classes of [9].…”

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confidence: 96%

Cellular Categories and Stable Independence

Lieberman¹,

Rosický²,

Vasey³

2022

J. symb. log.

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We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický.

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Accessible categories, set theory, and model theory: an invitation

Cited by 3 publications

References 37 publications

Formal model theory and higher topology

Formal model theory and higher topology

Formal Model Theory & Higher Topology

Cellular Categories and Stable Independence

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