Homotopy Canonicity for Cubical Type Theory

Leibniz International Proceedings in Informatics, Jun 2019

Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral.

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Homotopy Canonicity for Cubical Type Theory

F S C D Homotopy Canonicity for Cubical Type Theory Christian Sattler 0 1 2 0 Thierry Coquand Department of Computer Science and Engineering, University of Gothenburg , Sweden 1 Department of Computer Science and Engineering, University of Gothenburg , Sweden 2 Simon Huber Department of Computer Science and Engineering, University of Gothenburg , Sweden Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral. 2012 ACM Subject Classification Theory of computation ? Type theory; Theory of computation ? Proof theory; Theory of computation ? Computability Funding Simon Huber : I acknowledge the support of the Centre for Advanced Study (CAS) in Oslo, Norway, which funded and hosted the research project Homotopy Type Theory and Univalent Foundations during the academic year 2018/19. and phrases cubical type theory; univalence; canonicity; sconing; Artin glueing Introduction This paper is a contribution to the analysis of the computational content of the univalence axiom [34] (and higher inductive types). In previous work [2, 4, 6, 7, 23], various presheaf models of this axiom have been described in a constructive meta theory. In this formalism, the notion of fibrant type is stated as a refinement of the path lifting operation where one not only provides one of the endpoints but also a partial lift (for a suitable notion of partiality). This generalized form of path lifting operation is a way to state a homotopy extension property, which was recognized very early (see, e.g. [10]) as a key for an abstract development of algebraic topology. The axiom of univalence is then captured by a suitable equivalence extension operation (the ?glueing? operation), which expresses that we can extend a partially defined equivalence of a given total codomain to a total equivalence. These presheaf models suggest possible extensions of type theory where we manipulate higher dimensional objects [2, 6]. One can define a notion of reduction and prove canonicity for this extension [16]: any closed term of type N (natural number) is convertible to a numeral. There are however several non-canonical choices when defining the path lifting operation by induction on the type, which produce different notion of convertibility.1 A natural question is how essential these non-canonical choices are: can it be that a closed term of type N, defined without use of such non-canonical reduction rules, becomes convertible to 0 for one choice and 1 for another? 1 For instance, the definition of this operation for ?glue? types is different in [6] and [23]. ?(A, B)0(w) ?(A, B)0(w) fill?,b(u)0 Path(A, a0, a1)0(w) Glue(A, ? 7? (B, w))0v = = = = = ?(u : |A|)?(u0 : A0u).B0uu0(app(w, u)) ?(u0 : A0(fst(w))).B0(fst(w)) u0 (snd(w)) fill?,b(u0) Path?i.A0 i (ap(w,i)) a00 a01 Glue (A0(app(unglue, v))) [? 7? (B0v, (w0.1 v, . . .))] The main result of this paper, the homotopy canonicity theorem, implies that this cannot be the case. the value of a term is independent of these non-canonical choices. Homotopy canoncity states that, even without providing reduction rules for path lifting operations at type formers, we still have that any closed term of type N is path equal to a numeral. (We cannot hope to have convertibility anymore with these path lifting constant.) We can then see this numeral as the ?value? of the given term. Our proof of the homotopy canonicity can be seen as a proof-relevant extension of the reducibility or computability method, going back to the work of G?del [14] and Tait [32]. It is however best expressed in an algebraic setting. We first define a general notion of model, called cubical category with families, defined as a category with families [9] with certain special operations internal to presheaves over a category C (such as a cube category) with respect to the parameters of an interval I and an object of cofibrant propositions F. We describe the term model and how to re-interpret the cubical presheaf models as cubical categories with families. The computability method can then be expressed as a general operation (called ?sconing?) which applied to an arbitrary model M produces a new model M? with a strict morphism M? ? M. Homotopy canonicity is obtained by applying this general operation to the initial model, which we conjecture to be the term model. This construction associates to a (for simplicity, closed) type A a predicate A0 on the closed terms |A|, and each closed term u a proof u0 of A0u. The main rules in the closed case are summarized in Figure 1. (...truncated)


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Thierry Coquand, Simon Huber, Christian Sattler. Homotopy Canonicity for Cubical Type Theory, Leibniz International Proceedings in Informatics, 2019, pp. 11:1-11:23, 131, DOI: 10.4230/LIPIcs.FSCD.2019.11