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Article Dans Une Revue Journal of Logic and Computation Année : 2023

Projective classes as images of accessible functors

Résumé

We are dealing with projective classes (in short PC) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that PC(L_{\infty\lambda})=RPC(L_{\infty\lambda}) for any infinite cardinal \lambda, and that if \lambda, is singular, then PC(L_{\infty\lambda})=PC(L_{\infty\lambda^+}). If \lambda is regular, then a class of structures over a \lambda-ary vocabulary is PC(L_{\infty\lambda})-definable iff it is the image of a \lambda-continuous functor on a \lambda-accessible category. It follows that many PC classes of structures, previously known not to be closed under elementary equivalence over any L_{\infty\lambda}, are not co-PC over L_{\infty\infty}. Those classes arise from diverse contexts including convex l-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K_0-theory of rings, coordinatization of sectionally complemented modular lattices, and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is PC but not co-PC over L_{\infty\infty}. We also provide a negative solution to a problem, raised in~2011 by Gillibert and the author, asking whether essential surjectivity of a ``well-behaved'' functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.
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Dates et versions

hal-03580184 , version 1 (18-02-2022)
hal-03580184 , version 2 (12-04-2022)
hal-03580184 , version 3 (02-05-2022)

Identifiants

Citer

Friedrich Wehrung. Projective classes as images of accessible functors. Journal of Logic and Computation, 2023, 33 (1), pp.90-135. ⟨10.1093/logcom/exac043⟩. ⟨hal-03580184v3⟩
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