CJO - Abstract - Modular correspondence between dependent type theories and categories including pretopoi and topoi

Mathematical Structures in Computer Science (2005), 15 : 1089-1149 Cambridge University Press

Copyright © 2005 Cambridge University Press

doi:10.1017/S0960129505004962

Published online by Cambridge University Press 08 Dec 2005

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Mathematical Structures in Computer Science (2005), 15:6:1089-1149 Cambridge University Press
Copyright © 2005 Cambridge University Press
doi:10.1017/S0960129505004962

Paper

Modular correspondence between dependent type theories and categories including pretopoi and topoi

MARIA EMILIA MAIETTI ^a1
^a1 Dipartimento di Matematica Pura ed Applicata, Università di Padova, via G. Belzoni n. 7, I-35131 Padova, Italy Email: maietti@math.unipd.it

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Abstract

We present a modular correspondence between various categorical structures and their internal languages in terms of extensional dependent type theories à la Martin-Löf. Starting from lex categories, through regular ones, we provide internal languages of pretopoi and topoi and some variations of them, such as, for example, Heyting pretopoi.

With respect to the internal languages already known for some of these categories, such as topoi, the novelty of these calculi is that formulas corresponding to subobjects can be regained as particular types that are equipped with proof-terms according to the isomorphism ‘propositions as mono types’, which was invisible in previously described internal languages.

(Published Online December 8 2005)
(Received January 26 2004)
(Revised May 3 2005)