Categorical and algebraic aspects of Martin-löf type theory

Studia Logica 48 (3):299 - 317 (1989)
In the paper there are introduced and discussed the concepts of an indexed category with quantifications and a higher level indexed category to present an algebraic characterization of some version of Martin-Löf Type Theory. This characterization is given by specifying an additional equational structure of those indexed categories which are models of Martin-Löf Type Theory. One can consider the presented characterization as an essentially algebraic theory of categorical models of Martin-Löf Type Theory. The paper contains a construction of an indexed category with quantifications from terms and types of the language of Martin-Löf Type Theory given in the manner of Troelstra [11]. The paper contains also an inductive definition of a valuation of these terms and types in an indexed category with quantifications.
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DOI 10.1007/BF00370827
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M. Beeson (1982). Recursive Models for Constructive Set Theories. Annals of Mathematical Logic 23 (2-3):127-178.

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