Geodesic
A logic of orthogonality
Archivum mathematicum, Tome 42 (2006) no. 4, pp. 309-334 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A logic of orthogonality characterizes all “orthogonality consequences" of a given class $\Sigma $ of morphisms, i.e. those morphisms $s$ 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 Vopěnka’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 Jiří Rosický and the first two authors.
A logic of orthogonality characterizes all “orthogonality consequences" of a given class $\Sigma $ of morphisms, i.e. those morphisms $s$ 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 Vopěnka’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 Jiří Rosický and the first two authors.
Classification : 03C05, 03G30, 18C10, 18C35 
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Adámek, Jiří; Hébert, M.; Sousa, L. A logic of orthogonality. Archivum mathematicum, Tome 42 (2006) no. 4, pp. 309-334. http://geodesic.mathdoc.fr/item/ARM_2006_42_4_a0/

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