Path Spaces of Higher Inductive Types in Homotopy Type Theory

Kraus, Nicolai; Raumer, Jakob von

doi:10.1109/lics.2019.8785661

Cited by 11 publications

(15 citation statements)

References 32 publications

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“…The construction of ω and ϕ is essentially the same as before, using the version of Theorem 34 for pushouts available in [28]. For the relation (78), we can show the analogous to Lemmas 24 and 41.…”

Section: Applications In Homotopy Type Theorymentioning

confidence: 99%

A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

Kraus¹,

Raumer²

2021

Preprint

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Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types.This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higherdimensional rewriting.

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Section: Applications In Homotopy Type Theorymentioning

confidence: 99%

A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

Kraus¹,

Raumer²

2021

Preprint

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“…Theorem 9 (induction for coequaliser equality, [22]). Let a relation (∼) : A → A → U as before and a point a 0 : A be given.…”

Section: Definition 5 (Notation and Operations On Closures) We Have The Following Standard Operationsmentioning

confidence: 99%

“…The proof of Theorem 50 then proceeds as follows. The construction of ω and ϕ is essentially the same as before, using the version of Theorem 9 for pushouts available in [22]. For the relation (75), we can show the analogous to Lemmas 24 and 48.…”

Section: Applications Of Noetherian Cycle Inductionmentioning

confidence: 99%

Coherence via Well-Foundedness

Kraus

Raumer

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting -a natural operation which gives a new type for any binary relation on a type and, in order to be wellbehaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty.We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.

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“…As a simple but characteristic example, it implies function extensionality as a corollary: functions are identical when they are identical on all arguments [Uni13, §4.9]. Analogous extensionality principles for equality in coinductive types (e.g., [ACS15]) and quotients (e.g., [KvR19]) follow as well. In short, univalence regularizes the behavior of equality throughout type theory.…”

Section: Introductionmentioning

confidence: 99%

Internal Parametricity for Cubical Type Theory

Cavallo¹,

Harper²

2021

Logical Methods in Computer Science

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We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, observe how cubical equality regularizes parametric type theory, and examine the similarities and discrepancies between cubical and parametric type theory, which are closely related. We also abstract a formal interface to the computational interpretation and show that this also has a presheaf model.

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Path Spaces of Higher Inductive Types in Homotopy Type Theory

Cited by 11 publications

References 32 publications

A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

Coherence via Well-Foundedness

Internal Parametricity for Cubical Type Theory

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