Generalized sketches as a framework for completeness theorems. Part I

The concept of sketch is generalized. Morphisms of finite (generalized) sketches are used as sketch-entailments. A semantics and a deductive calculus of sketch-entailments are developed. A General Completeness Theorem (GCT) shows that the deductive calculus is adequate for the semantics. In each of...

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Bibliographic Details
Published inJournal of pure and applied algebra Vol. 115; no. 1; pp. 49 - 79
Main Author Makkai, M.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 14.02.1997
Subjects
03C99
03G30
18C10
03F99
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Summary:The concept of sketch is generalized. Morphisms of finite (generalized) sketches are used as sketch-entailments. A semantics and a deductive calculus of sketch-entailments are developed. A General Completeness Theorem (GCT) shows that the deductive calculus is adequate for the semantics. In each of a number of categories of sketches, a particular set of sketch-entailments is singled out as a set of axioms used to specify a particular kind of structured category. The specification yields an adequate proof-system to derive sketch-entailments valid in structured categories of the given kind. Classical, Tarski-type semantics is related to the sketch-semantics of the paper. Specific completeness theorems are given in the sketch-based formalism, and they are related to representation theorems of categorical logic, and known completeness theorems of logic.
ISSN:0022-4049
1873-1376
DOI:10.1016/S0022-4049(96)00007-2