Univalent higher categories via complete semi-segal types

Capriotti, Paolo and Kraus, Nicolai (2018) Univalent higher categories via complete semi-segal types. Proceedings of the ACM on Programming Languages, 2 (POPL). 44:1-44:29. ISSN 2475-1421

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Category theory in homotopy type theory is intricate as categorical laws can only be stated “up to homotopy”, and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.)solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories.

Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first (n + 3) levels of a semisimplicial type S, we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n-type. This is very similar to an earlier suggestion by Schreiber.

The definition of a univalent (1-) category by Ahrens et al. can easily be extended or restricted to the definition of a univalent n-category (more precisely, (n, 1)-category) for n ∈ {0, 1, 2}, and we show that the type of complete semi-Segal n-types is equivalent to the type of univalent n-categories in these cases. Thus, we believe that the notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category.

We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.

Item Type: Article
RIS ID: https://nottingham-repository.worktribe.com/output/907619
Keywords: Homotopy type theory, higher categories, complete Segal spaces
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Computer Science
Identification Number: https://doi.org/10.1145/3158132
Related URLs:
Depositing User: Kraus, Nicolai
Date Deposited: 29 Jan 2018 10:41
Last Modified: 04 May 2020 19:28
URI: https://eprints.nottingham.ac.uk/id/eprint/49374

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