Sets in homotopy type theory

TL;DR

This paper investigates the nature of sets within homotopy type theory, demonstrating that they form a $ ext{Pi}W$-pretopos and exploring conditions under which they form a topos, linking type theory with higher category theory.

Contribution

It proves that sets in homotopy type theory form a $ ext{Pi}W$-pretopos and analyzes the conditions for forming a topos, connecting homotopy type theory with $ ext{infty}$-topos theory.

Findings

Sets form a $ ext{Pi}W$-pretopos in homotopy type theory.

Both subobject and $0$-object classifiers exist for the universe of sets.

Under impredicative propositional resizing, the sets form a topos.

Abstract

Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak $\infty$-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those `discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of `elementary' $\infty$-toposes. We prove that sets in homotopy type theory form a $\Pi W$-pretopos. This is similar to the fact that the $0$-truncation of an $\infty$-topos is a topos. We show that both a subobject classifier and a $0$-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover, the $0$-object classifier for sets is a function between $1$-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.