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Categories with families, FOLDS and logic enriched type theory
Stockholm University, Faculty of Science, Department of Mathematics.
2016 (English)Manuscript (preprint) (Other academic)
Abstract [en]

Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Lö type theory. Makkai's first-order logic with dependent sorts (FOLDS) is an example of a so-called logic enriched type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL) fit in this semantical framework. A soundness and completeness theorem is proved for such a logic. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Some applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle.

Place, publisher, year, edition, pages
2016. , 99 p.
Keyword [en]
Intuitionistic first-order logic, dependent types, categorical logic, models of type theory
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
URN: urn:nbn:se:su:diva-129963OAI: oai:DiVA.org:su-129963DiVA: diva2:926427
Projects
Konstruktiva och kategoriteoretiska grundvalar för matematik
Funder
Swedish Research Council, 2015-03835
Available from: 2016-05-06 Created: 2016-05-06 Last updated: 2016-05-12

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arXiv:1605.01586

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