Homotopy Type Theory: The Logic of Space

Shulman, Michael

doi:10.1017/9781108854429.009

Cited by 18 publications

(29 citation statements)

References 48 publications

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“…On the other hand, complementary to what we have shown here, there is an ongoing research program, involving the application of methods from homotopy type theory and infinity-category theory, that seeks to formalize, among other things, the mathematical foundations of synthetic spatial structures (in particular, cohesive topological and geometric spaces) [30] [25] [26]. Constructions of synthetic geometry are also of great relevance for the foundations of physics.…”

Section: Implications and Outlookmentioning

confidence: 99%

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Arsiwalla¹,

Gorard²,

Elshatlawy³

2021

Preprint

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We investigate the algebraic and compositional properties of multiway (non-deterministic) abstract rewriting systems, which are the archetypical structures underlying the formalism of the so-called Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where these homotopic maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order n may naturally be formalized as an n-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ∞-groupoid. Via Grothendieck's homotopy hypothesis, this ∞-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of multiway rewriting systems may potentially be used for making formal connections to homotopy spaces upon which models of physics can be instantiated.

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Section: Implications and Outlookmentioning

confidence: 99%

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Arsiwalla¹,

Gorard²,

Elshatlawy³

2021

Preprint

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“…As mentioned in the introduction, our work borrows heavily from recent advances in the foundations of mathematics, particularly homotopy type theory [75] and synthetic geometry [85], [86], [82]. Our long-term goals with respect to homotopy type theory and higher categorical structures to physics are: (i) to explore higher symmetries and spaces, that cannot readily be captured by current methods; and (ii) to seek a constructivist foundation for theories of physics in much the same way that such a foundation is proving fruitful for mathematics itself.…”

Section: Relation To Other Workmentioning

confidence: 99%

“…The relevant type constructors for this are the dependent coproduct type and a monoid type. For the former, one can simply invoke the usual coproduct constructor (this has been nicely detailed in [86]); whereas, the latter, would be similar to the monoids we constructed for the string substitution multiway above. Each E l is a monoid over a finite generating subset of N with the identity element simply being the null element φ.…”

Section: (Hyper)graph Rewriting Multiway Systemsmentioning

confidence: 99%

“…Following the above remark, note that the existence (or lack of it thereof) and construction of cohesive structures on limiting rulial multiway systems will certainly depend on the rules used to construct that multiway graph, which in turn determine the growth of that multiway system (see [95] for a classification on multiway growth functions). In [82], [86] two concrete examples of cohesive structures have been reported: those constructed within a topos of convergent sequences or what are referred to as 'consequential spaces' {N ∞ }; and those constructed within a category of abstract co-ordinate charts {R n } (as types, not as sets or topological spaces) for all n ∈ N. In future work, it will be interesting to investigate other realizations of cohesive structures, using multiway systems.…”

Section: The ∞-Limit Of Multiway Rewriting Systems As ∞-Groupoidsmentioning

confidence: 99%

“…Furthermore, we show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an (∞, 1)-topos. This correspondence to spaces and higher structures offers a formal understanding of how spatial structures relevant to theories of physics can be shown to originate from abstract combinatorial constructions [6], [9], [86], [16].…”

Section: Introductionmentioning

confidence: 99%

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Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

Arsiwalla¹,

Gorard²

2021

Preprint

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How do spaces in physics emerge from pregeometric discrete building blocks governed by computational rules? To address this question we investigate non-deterministic rewriting systems (so-called multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation of the model, we motivate how spatial structures are functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopies are constructed from rewriting rules. These correspond to morphisms of an n-fold category. Subsequently, the n → ∞ limit of the Wolfram model rulial multiway system is identified as an ∞-groupoid, with the latter being relevant in light of Grothendieck's homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an (∞, 1)-topos. This correspondence to higher categorical structures potentially offers a new way to understand how the kinds of spaces relevant to physics result from pregeometric combinatorial models. The key issue we have addressed in this work is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it removes the need to make ad hoc assumptions about geometric attributes of a model such as an a priori background or any pre-assigned geometric data. Instead, geometry is inherited functorially from higher structures. This can be particularly useful for formally justifying different choices of underlying spacetime discretization schemes adopted by various models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions closely corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.

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Steps Toward a Philosophy for Mathematicians

Fenstad

2022

Math Intelligencer

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Homotopy Type Theory: The Logic of Space

Cited by 18 publications

References 48 publications

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

Steps Toward a Philosophy for Mathematicians

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