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Preprint Number 1367

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1367. Michael Lieberman, Jiří Rosický, and Sebastien Vasey
Forking independence from the categorical point of view
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Submission date: 26 January 2018

Abstract:

We develop the theory of forking-like independence in accessible categories. To do so, we present an axiomatic definition of what we call a stable independence notion and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory.
We then show that any coregular locally presentable category with effective unions admits a forking-like independence notion. This includes the cases of Grothendieck toposes and Grothendieck abelian categories.
We also give conditions for existence and canonicity of stable independence notions. Specifically, an accessible category with directed colimits whose morphisms are monomorphisms will have at most one stable independence notion. Moreover, assuming a large cardinal axiom, a cofinal full subcategory will have a stable independence notion if and only if a certain order property fails. This establishes, in particular, a category-theoretic characterization of stability (in the model-theoretic sense) for accessible categories.

Mathematics Subject Classification: 03C45 (Primary), 18C35, 03C48, 03C52, 03C55, 03C75, 03E55 (Secondary)

Keywords and phrases:

Full text arXiv 1801.09001: pdf, ps.


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