Constructions with non-recursive higher inductive types

Kraus, Nicolai (2016) Constructions with non-recursive higher inductive types. In: Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), July 5–8, 2016, New York City, USA. (In Press)

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Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say that a HIT H is non-recursive if its constructors do not quantify over elements or paths in H. The advantage of non-recursive HITs is that their elimination principles are easier to apply than those of general HITs.

It is an open question which classes of HITs can be encoded as non-recursive HITs. One result of this paper is the construction of the propositional truncation via a sequence of approximations, yielding a representation as a non-recursive HIT. Compared to a related construction by van Doorn, ours has the advantage that the connectedness level increases in each step, yielding simplified elimination principles into n-types. As the elimination principle of our sequence has strictly lower requirements, we can then prove a similar result for van Doorn’s construction. We further derive general elimination principles of higher truncations (say, k-truncations) into n-types, generalizing a previous result by Capriotti et al. which considered the case n=k+1.

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Keywords: Homotopy type theory
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Computer Science
Depositing User: Kraus, Nicolai
Date Deposited: 28 Apr 2016 13:14
Last Modified: 04 May 2020 20:05

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