10 research outputs found
Unstable independence from the categorical point of view
We give a category-theoretic construction of simple and NSOP-like
independence relations in locally finitely presentable categories, and in the
more general locally finitely multipresentable categories. We do so by
identifying properties of a class of monomorphisms such that the
pullback squares consisting of morphisms in form the desired
independence relation. This generalizes the category-theoretic construction of
stable independence relations using effective unions or cellular squares by M.
Lieberman, S. Vasey and the second author to the unstable setting.Comment: 32 page
Structural Logic and Abstract Elementary Classes with Intersection
We give a syntactic characterization of abstract elementary classes (AECs)
closed under intersections using a new logic with a quantifier for isomorphism
types that we call structural logic: we prove that AECs with intersections
correspond to classes of models of a universal theory in structural logic. This
generalizes Tarski's syntactic characterization of universal classes. As a
corollary, we obtain that any AEC with countable L\"owenheim-Skolem number is
axiomatizable in , where is the quantifier
"there exists uncountably many".Comment: 14 page