Categorical and algebraic aspects of Martin-Löf Type Theory

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.

[1]  Robert Paré,et al.  Abstract families and the adjoint functor theorems , 1978 .

[2]  Michael Beeson,et al.  Recursive models for constructive set theories , 1982, Ann. Math. Log..

[3]  S. Lane.  Categories for the Working Mathematician , 1971 .

[4]  P. Martin-Löf.  Constructive mathematics and computer programming , 1984 .

[5]  Anne Sjerp Troelstra,et al.  On the Syntax of Martin-Löf's Type Theories , 1987, Theor. Comput. Sci..

[6]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[7]  R. Seely,et al.  Locally cartesian closed categories and type theory , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  John Cartmell,et al.  Generalised algebraic theories and contextual categories , 1986, Ann. Pure Appl. Log..

[9]  Peter Freyd,et al.  Aspects of topoi , 1972, Bulletin of the Australian Mathematical Society.