# Towards a directed homotopy type theory

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-Löf type theory which is roughly analogous to the identity type former originally introduced by Martin-Löf. 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-Löf's identity types.

## Authors

• 7 publications
02/27/2019

### From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

Cube categories are used to encode higher-dimensional structures. They h...
03/27/2018

### Univalent polymorphism

We show that Martin Hyland's effective topos can be exhibited as the hom...
07/13/2021

### From Identity to Difference: A Quantitative Interpretation of the Identity Type

We explore a quantitative interpretation of 2-dimensional intuitionistic...
03/03/2020

### A Cubical Language for Bishop Sets

We present XTT, a version of Cartesian cubical type theory specialized f...
05/19/2018

### A Compositional Approach to Network Algorithms

We present elements of a typing theory for flow networks, where "types",...
07/16/2020

### A Type Theory for Strictly Unital ∞-Categories

We present a type theory for strictly unital ∞-categories, in which a te...
02/26/2022

### A Synthetic Perspective on (∞,1)-Category Theory: Fibrational and Semantic Aspects

Reasoning about weak higher categorical structures constitutes a challen...
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