Data structures for quasistrict higher categories

Bar, Krzysztof; Vicary, Jamie

doi:10.1109/lics.2017.8005147

Cited by 12 publications

(22 citation statements)

References 39 publications

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“…For our purposes this will be omitted to simplify the presentation, as our results are applicable to any signature. This encoding scheme is essentially identical to that used by the proof assistant Globular [BV16], although the result in this section is new, and is not implied by the existing literature. This encoding scheme serves as a formal combinatorial foundation for our results, although we will build most of our arguments at a more intuitive level with the corresponding graphical diagrams.…”

Section: Monoidal Categories and String Diagramsmentioning

confidence: 99%

Normalization for planar string diagrams and a quadratic equivalence algorithm

Delpeuch¹,

Vicary²

2022

Logical Methods in Computer Science

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In the graphical calculus of planar string diagrams, equality is generated by exchange moves, which swap the heights of adjacent vertices. We show that left- and right-handed exchanges each give strongly normalizing rewrite strategies for connected string diagrams. We use this result to give a linear-time solution to the equivalence problem in the connected case, and a quadratic solution in the general case. We also give a stronger proof of the Joyal-Street coherence theorem, settling Selinger's conjecture on recumbent isotopy.

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Section: Monoidal Categories and String Diagramsmentioning

confidence: 99%

Normalization for planar string diagrams and a quadratic equivalence algorithm

Delpeuch¹,

Vicary²

2022

Logical Methods in Computer Science

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“…Moreover, contrarily to strict higher categories, their cells can be easily described by normal forms, making them amenable to computations. This notion was used to give several definitions of semi-strict higher categories [4] and is the underlying structure of the Globular tool for higher categories [3]. Premises of it can be found in the work of Street [28] and Makkai [22].…”

Section: Precategoriesmentioning

confidence: 99%

“…The notion of precategory is a generalization of the one of sesquicategory, whose use has already been advocated by Street in the context of rewriting [28]. The interest in those has also been renewed recently, because they are at the heart of the graphical proof-assistant Globular [3,4]. Gray categories are particular 3-precategories equipped with exchange 3-cells satisfying suitable axioms.…”

Section: Introductionmentioning

confidence: 99%

Rewriting in Gray categories with applications to coherence

Forest¹,

Mimram²

2021

Preprint

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Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which include Gray categories as particular cases, and are adapted to mechanized computations. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed. We also extend Squier's theorem to our context, showing that a convergent rewriting system is coherent, which means that any two parallel 3-cells are necessarily equal. This allows us to prove coherence results for several well-known structures in the context of Gray categories: monoids, adjunctions, Frobenius monoids.

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“…The theory of ANCs can be seen as a development into arbitrary dimension of the theory of quasistrict 4-categories of Bar and Vicary [8], implemented as the proof assistant Globular by Bar, Kissinger and Vicary [9]. That proof assistant had a restricted notion of homotopy construction, limited fundamentally to dimension 4, and could not even in principle be generalized to arbitrary dimension, where our results apply.…”

Section: Related Workmentioning

confidence: 99%

“…The length of the bottom-right zigzag, and its incoming monotone maps, are determined by taking a pushout in ∆. The regular objects of the bottomright zigzag are completely determined by the incoming maps, and the singular objects8 Recall Definition 17 of concatenation of zigzags and their maps.…”

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confidence: 99%

High-level methods for homotopy construction in associative n-categories

Reutter

Vicary

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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A combinatorial theory of associative n-categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a notion of 'homotopy' with a natural geometrical interpretation. Such a theory has the potential to serve as an attractive foundation for a computer proof assistant for higher category theory, since it allows composites to be uniquely described, and relieves proofs from the bureaucracy of associators, unitors and their coherence. However, this basic theory lacks a high-level way to construct homotopies, which would be intractable to build directly in complex situations; it is not therefore immediately amenable to implementation.We tackle this problem by describing a 'contraction' operation, which algorithmically constructs complex homotopies that reduce the lengths of composite terms. This contraction procedure allows building of nontrivial proofs by repeatedly contracting subterms, and also allows the contraction of those proofs themselves, yielding in some cases single-step witnesses for complex homotopies. We prove correctness of this procedure by showing that it lifts connected colimits from a base category to a category of zigzags, a procedure which is then iterated to yield a contraction mechanism in any dimension. We also present homotopy.io, an online proof assistant that implements the theory of associative n-categories, and use it to construct a range of examples that illustrate this new contraction mechanism. 12 1 arXiv:1902.03831v1 [math.CT]

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Data structures for quasistrict higher categories

Cited by 12 publications

References 39 publications

Normalization for planar string diagrams and a quadratic equivalence algorithm

Normalization for planar string diagrams and a quadratic equivalence algorithm

Rewriting in Gray categories with applications to coherence

High-level methods for homotopy construction in associative n-categories

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