Coherence and strictification for self-similarity

Hines, Peter

doi:10.1007/s40062-016-0154-y

Cited by 6 publications

(17 citation statements)

References 31 publications

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“…The and isomorphisms are then mutually inverse bĳections exhibiting ⊗ . However, these are unique up to unique isomorphism [31], so our result follows. This convergence of the elements of reflexivity is particularly relevant for compact closed categories arising from the Int or GoI construction, where all self-dual objects are by construction isomorphic to some strictly self-dual object.…”

Section: Remark 14 the Best-known Examples Of Daggers On Compact Clos...mentioning

confidence: 59%

“…An object ∈ (C) is called self-similar or pseudo-idempotent when it satisfies ⊗ . The isomorphisms exhibiting this self-similarity are unique up to unique isomorphism [31], and commonly referred to as the code and decode arrows ⊳ ∈ C( ⊗ , ) and ⊲ ∈ C( , ⊗ ) respectively.…”

Section: Reflexive Objects Of Compact Closed Categoriesmentioning

confidence: 99%

“…It is now inevitable that we consider how self-similarity may be strictified. We are fortunate to be able to rely on a pre-existing coherence theorem and strictification procedure [31]. However, somewhat inconveniently for our aim of finding a monoidal strictification of reflexivity, this firmly and unavoidably lives within the theory of semi-monoidal categories.…”

Section: Strict Self-similaritymentioning

confidence: 99%

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On strict extensional reflexivity in compact closed categories

Hines¹

2022

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This article has two related aims. The first is to study the categorical setting of Abramsky, Haghverdi, & Scott's untyped linear combinatory algebras [6], and the second is to relate this to the much more recent work by Abramsky & Heunen on Frobenius algebras in the infinitary setting [7]. The key to this is (extensional) reflexivity (i.e. the property of an object being isomorphic to its own internal hom[ → ]). We first characterise extensionally reflexive objects in compact closed categories, then consider when & how this property may be 'strictified' -how we may give a monoidally equivalent category where the isomorphisms exhibiting reflexivity are in fact identity arrows. This results in small two-object compact closed categories consisting of a unit object and a single (non-unit) strictly reflexive object. We then move on to studying the endomorphism monoids of such objects from an algebraic rather than logical or categorical viewpoint. We demonstrate that these necessarily contain an interesting inverse monoid that may be thought of as Richard Thompson's iconic group F together with the equally iconic bicyclic monoid B of semigroup theory, with non-trivial interactions between the two derived from the Frobenius algebra identity -and claim this as a particularly significant example of the (unitless) Frobenius algebras of [7]. We first develop the theory from a purely theoretical point of view, then move on to develop concrete examples, based on the algebra and category theory behind [21,22,6]. The concrete examples we give are based on the traced monoidal category of partial injections, and reflexive objects in the compact closed category that results from applying the Int or GoI construction. We then give compact closed categories, monoidally equivalent to compact closed subcategories of Int(pInj), where this reflexivity is exhibited by identity arrows, and show how the above algebraic structures (Thompson's F , the bicyclic monoid, and Frobenius algebras) arise in a fundamental manner. This work is of course !( ) dedicated to Samson Abramsky. Historical backgroundThe starting point for this chapter is Girard's Geometry of Interaction programin particular the first two parts. It is by now well-established that compact closed 1 categories model key aspects of this (see [6,25] for a good account), and this observation motivated the name of Abramsky's categorical GoI construction [1] (see Section 7).However, this immediately poses an interesting puzzle. In [21], Girard makes the rather cryptic comment that his system 'forgets types', even though the stated aim was to produce a model of the polymorphically typed System F , rather than a purely untyped system. The explanation seems to be that the GoI system moves from a rigidly typed system to an entirely untyped system, in order to build a more flexible (polymorphic) type system on top of this . The claim that there is an untyped logical system at the core of Girard's GOI was borne out in Abramsky, Haghverdi, and Scott's paper [6] that gave an untyped combinat...

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Section: Remark 14 the Best-known Examples Of Daggers On Compact Clos...mentioning

confidence: 59%

Section: Reflexive Objects Of Compact Closed Categoriesmentioning

confidence: 99%

Section: Strict Self-similaritymentioning

confidence: 99%

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On strict extensional reflexivity in compact closed categories

Hines¹

2022

Preprint

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“…We refer to [20,21] for the theory of semi-monoidal categories, and [22,23] for the monoid-theoretic setting. The following is well-established : Theorem 16.…”

Section: An Interpretation As Categorical Coherencementioning

confidence: 99%

“…A historical account together with a proof (with no claim to originality) is given in [24]. Key contributions include, but are not restricted to, [10,23,25,9,16]; not all of these are phrased categorically.…”

Section: An Interpretation As Categorical Coherencementioning

confidence: 99%

From a conjecture of Collatz to Thompson's group F, via a conjunction of Girard

Hines¹

2022

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The famous 3x + 1 problem of L. Collatz needs no introduction; however, this paper concerns a lesser-known, but similarly unresolved, precursor problem : the Original Collatz Conjecture, or OCC [1].We demonstrate that the core arithmetic operator from the OCC, when combined with a conjunction of J.-Y. Girard from his Geometry of Interaction system [2, 3], leads to a realisation of R. Thompson's group F as congruential functions, in the sense of J. Conway [4].We also give the underlying category theory that accounts for this, and describe the core operator from the OCC as a canonical coherence isomorphism.

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On Strictifying Extensional Reflexivity in Compact Closed Categories

Hines

2023

Outstanding Contributions to Logic

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Coherence and strictification for self-similarity

Cited by 6 publications

References 31 publications

On strict extensional reflexivity in compact closed categories

On strict extensional reflexivity in compact closed categories

From a conjecture of Collatz to Thompson's group F, via a conjunction of Girard

On Strictifying Extensional Reflexivity in Compact Closed Categories

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