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A Logic of Orthogonality , M. Hebert and L. Sousa

Summary: A Logic of Orthogonality
J. Ad´amek
, M. H´ebert and L. Sousa
This paper was inspired by the hard-to-believe
fact that Jir´i Rosick´y is getting sixty. We are
happy to dedicate our paper to Jirka on the oc-
casion of his birthday.
A logic of orthogonality characterizes all "orthogonality consequences" of a
given class of morphisms, i.e. those morphisms s such that every object or-
thogonal to is also orthogonal to s. A simple four-rule deduction system is
formulated which is sound in every cocomplete category. In locally presentable
categories we prove that the deduction system is also complete (a) for all classes
of morphisms such that all members except a set are regular epimorphisms and
(b) for all classes , without restriction, under the set-theoretical assumption that
Vopenka's Principle holds. For finitary morphisms, i.e. morphisms with finitely
presentable domains and codomains, an appropriate finitary logic is presented,
and proved to be sound and complete; here the proof follows immediately from
previous joint results of Jir´i Rosick´y and the first two authors.
1 Introduction


Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig


Collections: Computer Technologies and Information Sciences