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Categorical and Algebraic Aspects of Martin-Löf Type Theory

Adam Obtułowicz
Studia Logica: An International Journal for Symbolic Logic
Vol. 48, No. 3 (1989), pp. 299-317
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015445
Page Count: 19
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Categorical and Algebraic Aspects of Martin-Löf Type Theory
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Abstract

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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