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Categories with families, FOLDS and logic enriched type theory

Authors
  • Palmgren, Erik
Type
Preprint
Publication Date
May 05, 2016
Submission Date
May 05, 2016
Identifiers
arXiv ID: 1605.01586
Source
arXiv
License
Yellow
External links

Abstract

Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-L\"of 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.

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