Sets in homotopy type theory

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

The Sierpinski Object in the Scott Realizability Topos

We study the Sierpinski object $\Sigma$ in the realizability topos based on Scott's graph model of the $\lambda$-calculus. Our starting observation is that the object of realizers in this topos is the exponential $\Sigma ^N$, where $N$ is the natural numbers object. We define order-discrete objects by orthogonality to $\Sigma$. We show that the order-discrete objects form a reflective subcategory of the topos, and that many fundamental objects in higher-type arithmetic are order-discrete. Building on work by Lietz, we give some new results regarding the internal logic of the topos. Then we consider $\Sigma$ as a dominance; we explicitly construct the lift functor and characterize $\Sigma$-subobjects. Contrary to our expectations the dominance $\Sigma$ is not closed under unions. In the last section we build a model for homotopy theory, where the order-discrete objects are exactly those objects which only have constant paths

Internal Universes in Models of Homotopy Type Theory

We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory

Partial functions and recursion in univalent type theory

We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability theory. We begin with a treatment of partial functions, using the notion of dominance, which is used in synthetic domain theory to discuss classes of partial maps. We relate this and other ideas from synthetic domain theory to other approaches to partiality in type theory. We show that the notion of dominance is difficult to apply in our setting: the set of �0 1 propositions investigated by Rosolini form a dominance precisely if a weak, but nevertheless unprovable, choice principle holds. To get around this problem, we suggest an alternative notion of partial function we call disciplined maps. In the presence of countable choice, this notion coincides with Rosolini’s. Using a general notion of partial function,we take the first steps in constructive computability theory. We do this both with computability as structure, where we have direct access to programs; and with computability as property, where we must work in a program-invariant way. We demonstrate the difference between these two approaches by showing how these approaches relate to facts about computability theory arising from topos-theoretic and typetheoretic concerns. Finally, we tie the two threads together: assuming countable choice and that all total functions N - N are computable (both of which hold in the effective topos), the Rosolini partial functions, the disciplined maps, and the computable partial functions all coincide. We observe, however, that the class of all partial functions includes non-computable partial functions