Towards a directed homotopy type theory

TL;DR

This paper introduces a directed homotopy type theory with a new homomorphism type former, interpreting it in the category of small categories to model morphisms and directed paths, advancing reasoning about higher categories and concurrency.

Contribution

It proposes a novel homomorphism type former for Martin-Löf type theory, capturing directed morphisms and paths, and provides an interpretation in Cat linking type theory with category theory.

Findings

Homomorphism types interpreted as morphisms in Cat

Homomorphism types serve as directed analogs of identity types

Framework supports reasoning about higher categories and concurrency

Abstract

In this paper, we present a directed homotopy type theory for reasoning synthetically about (higher) categories, directed homotopy theory, and its applications to concurrency. We specify a new `homomorphism' type former for Martin-L\"of type theory which is roughly analogous to the identity type former originally introduced by Martin-L\"of. The homomorphism type former is meant to capture the notions of morphism (from the theory of categories) and directed path (from directed homotopy theory) just as the identity type former is known to capture the notions of isomorphism (from the theory of groupoids) and path (from homotopy theory). Our main result is an interpretation of these homomorphism types into Cat, the category of small categories. There, the interpretation of each homomorphism type hom(a,b) is indeed the set of morphisms between the objects a and b of a category C. We end the paper with an analysis of the interpretation in Cat with which we argue that our homomorphism types are indeed the directed version of Martin-L\"of's identity types.