On Equality of Objects in Categories in Constructive Type Theory

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. 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. In fact, for every type A, there is an E-groupoid A which cannot be so extended. We also introduce the notion of an H-category, a variant of category, which makes it easy to compare to the notion of "univalent" category proposed in Univalent Type Theory [9]. When formalizing mathematical structures in constructive type theory it is common to interpret the notion of set as a type together with an equivalence relation, and the notion of function between sets as a function or operation that preserves the equivalence relations. Such functions are called extensional functions. This way of interpreting sets was adopted in Bishop’s seminal book [3] on constructive analysis from 1967. In type theory literature such sets are called setoids. Formally a setoid X = (|X|,=X) consists of a type |X| together with a binary relation =X , and a proof object for =X being an equivalence relation. An extensional function between setoids f : X // Y consists of a type-theoretic function |f | : |X| // |Y |, and a proof that f respects the equivalence relations, i.e. |f |(x) =Y |f |(u) whenever x =X u. One writes x : X for x : |X|, and f(x) for |f |(x) to simplify notation. Every type A comes with a minimal equivalence relation IA(·, ·), the so-called identity type for A. When the type can be inferred we write a . = b for IA(a, b). The principle of Uniqueness of Identity Proofs (UIP) for a type A states that (UIPA) (∀a, b : A)(∀p, q : a . = b)p . = q. This principle is not assumed in basic type theory, but can be proved for types A where IA(·, ·) is a decidable relation (Hedberg’s Theorem [9]). Date: August 5, 2017. 1