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arXiv · 1708.01924

On equality of objects in categories in constructive type theory

Erik Palmgren
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Abstract

In this note we remark on the problem of equality of objects in categories formalized in Martin-Löf's constructive type theory. A standard notion of category in this system is E-category, where no such equality is specified. The main observation here is that there is no general extension of E-categories to categories with equality on objects, unless the principle Uniqueness of Identity Proofs (UIP) holds. We also introduce the notion of an H-category, a variant of category with equality on objects, which makes it easy to compare to the notion of univalent category proposed for Univalent Type Theory by Ahrens, Kapulkin and Shulman.

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Erik Palmgren. 2017-08-06. On equality of objects in categories in constructive type theory. https://doi.org/10.4230/lipics.types.2017.7

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