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A logic of orthogonality

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J. Adamek, M. Hebert, L. Sousa
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** Address.**
Technical University of Braunschweig, Germany

The American University of Cairo, Egypt

Technical University of Viseu, Portugal

** E-mail:**
J.Adamek@tu-bs.de,
mhebert@aucegypt.edu,
sousa@mat.estv.ipv.pt

**Abstract.**
A logic of orthogonality characterizes all ``orthogonality consequences" of a given class $\Sigma$ of morphisms, i.e. those morphism $s$ such that every object orthogonal to $\Sigma$ 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 $\Sigma$ of morphisms such that all members except a set are regular epimorphisms and (b) for all classes $\Sigma$, 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 point results of Jiri Rosicky and the first two authors.