Classifying topoi in synthetic guarded domain theory

Several different topoi have played an important role in the development and applications of synthetic guarded domain theory (SGDT), a new kind of synthetic domain theory that abstracts the concept of guarded recursion frequently employed in the semantics of programming languages. In order to unify the accounts of guarded recursion and coinduction, several authors have enriched SGDT with multiple "clocks" parameterizing different time-streams, leading to more complex and difficult to understand topos models. Until now these topoi have been understood very concretely qua categories of presheaves, and the logico-geometrical question of what theories these topoi classify has remained open. We show that several important topos models of SGDT classify very simple geometric theories, and that the passage to various forms of multi-clock guarded recursion can be rephrased more compositionally in terms of the lower bagtopos construction of Vickers and variations thereon due to Johnstone. We contribute to the consolidation of SGDT by isolating the universal property of multi-clock guarded recursion as a modular construction that applies to any topos model of single-clock guarded recursion.Comment: To appear in the proceedings of the 38th International Conference on Mathematical Foundations of Programming Semantics (MFPS 2022

Natural models of homotopy type theory

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums, dependent products, and intensional identity types, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: they should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.Comment: 51 page

A logical study of some 2-categorical aspects of topos theory

There are two well-known topos-theoretic models of point-free generalized spaces: the original Grothendieck toposes (relative to classical sets), and a relativized version (relative to a chosen elementary topos $S$ with a natural number object) in which the generalized spaces are the bounded geometric morphisms from an elementary topos $E$ to $S$, and they form a 2-category $BTop/S$. However, often it is not clear what a preferred choice for the base $S$ should be. In this work, we review and further investigate a third model of generalized spaces, based on the 2-category $Con$ of ‘contexts for Arithmetic Universes (AUs)’ presented by AU-sketches which originally appeared in Vickers’ work in [Vic19] and [Vic17]. We show how to use the AU techniques to get simple proofs of conceptually stronger, base-independent, and predicative (op)fibration results in $ETop$, the 2-category of elementary toposes equipped with a natural number object, and arbitrary geometric morphisms. In particular, we relate the strict Chevalley fibrations, used to define fibrations of AU-contexts, to non-strict Johnstone fibrations, used to define fibrations of toposes. Our approach brings to light the close connection of (op)fibration of toposes, conceived as generalized spaces, with topological properties. For example, every local homeomorphism is an opfibration and every entire map (i.e. fibrewise Stone) is a fibration

Variations on the bagdomain theme

AbstractThe notion of bagdomain was first introduced by Vickers, (1992) and further studied by the present author in (Johnstone, 1992). In these papers, attention was focused on one particular version of the bagdomain construction, the “bag” analogue of the lower (Hoare) powerdomain; but there are many other possibilities. The purpose of the present paper is to introduce some of these possibilities and to describe their basic properties, using the theory of fibrations and partial products developed in (Johnstone, 1993)