Abstract
In this paper, we present a directed homotopy type theory for reasoning synthetically about (higher) categories and directed homotopy theory. 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 homC(a,b) is indeed the set of morphisms between the objects a and b of the 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
Highlights
Martin-Lof type theory, together with its identity type, is often described as a synthetic theory of higher groupoids
We end the paper with an analysis of the interpretation in Cat with which we argue that our homomorphism types are the directed version of Martin-Lof’s identity types
This rich structure of the identity type was first discovered by Hofmann and Streicher in their disproof of the uniqueness of identity proofs [HS96]
Summary
Martin-Lof type theory, together with its identity type, is often described as a synthetic theory of higher groupoids. The full extent of this structure was later made explicit in Voevodsky’s interpretation of Martin-Lof type theory with the identity type (amongst other types) into the category of Kan complexes, the objects of which represent spaces or ∞-groupoids [KL12]. In [Nor17], the author has shown that in any finitely complete category, models of Martin-Lof’s identity type are in correspondence with weak factorization systems with two stability properties Such results describe a fascinating perspective on the theory of higher groupoids. One might hope that such a type theory would shed light on the less well understood theories of higher categories and directed spaces. Both of these theories have a notion of directed paths at their core, and for both of these theories, there are many competing concrete models.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
