Equational Properties of Fixed Point Operations in Cartesian Categories: An Overview

Ésik, Zoltán

doi:10.1007/978-3-662-48057-1_2

Cited by 7 publications

(11 citation statements)

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“…An iteration category [15] is a cartesian category with a parametrized fixed point operator that behaves in a canonical way. The definition of an iteration category came out of the observation that the parametrized fixed point operator in a host of concrete categories (notably DCPO) satisfy the same identities.…”

Section: Parametrized Fixed Points and Iteration Categoriesmentioning

confidence: 99%

“…The definition of an iteration category came out of the observation that the parametrized fixed point operator in a host of concrete categories (notably DCPO) satisfy the same identities. This lead to an elegant semantic characterization of iteration categories, due to [15]. Definition 2.…”

Section: Parametrized Fixed Points and Iteration Categoriesmentioning

confidence: 99%

“…In recent years, dagger categories have been used to capture aspects of inversion in both reversible [27,28,30] and quantum [2,35,12] computing. Likewise, domain theory and categories enriched in domains (see, e.g., [3,14,15,4,6,38]) have been successful since their inception in modelling both recursive functions and data types in programming via fixed points.…”

Section: Introductionmentioning

confidence: 99%

“…Though familiarity with basic category theory, including monoidal categories, is assumed, we recall here some basic concepts relating to dagger categories, (DCPO)-enriched categories, iteration categories, and involutive monoidal categories [25,7]. The material is only covered here briefly, but can be found in much more detail in the numerous texts on dagger category theory (see, e.g., [35,2,20,31]), enriched category theory (for which [33] is the standard text), and domain theory and iteration categories (see, e.g., [3,15]).…”

Section: Introductionmentioning

confidence: 99%

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Inversion, Iteration, and the Art of Dual Wielding

Kaarsgaard

2019

Preprint

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The humble † ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact. In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagger structure that is well behaved with respect to the enrichment, and show that such a structure leads to pleasant inversion properties of the fixed points that arise as a result. Notably, such a structure guarantees the existence of fixed point adjoints, which we show are intimately related to the conjugates arising from a canonical involutive monoidal structure in the enrichment. Finally, we relate the results to applications in the design and semantics of reversible programming languages.Keywords: reversible computing • dagger categories • iteration categories • domain theory • enriched categories ⋆ The author would like to thank Martti Karvonen, Mathys Rennela, and Robert Glück for their useful comments, corrections, and suggestions; and to acknowledge the support given by COST Action IC1405 Reversible computation: Extending horizons of computing.

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Section: Parametrized Fixed Points and Iteration Categoriesmentioning

confidence: 99%

Section: Parametrized Fixed Points and Iteration Categoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inversion, Iteration, and the Art of Dual Wielding

Kaarsgaard

2019

Preprint

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“…Every Cartesian category C is guarded under Hom π pX, Y q " HompX, Y q (making every morphism guarded). Then C has a guarded Conway recursion operator iff C is a Conway category [13], i.e. models standard total recursion.…”

Section: Guardedness Via Guarded Idealsmentioning

confidence: 99%

Guarded Traced Categories

Goncharov

Schröder

2018

Lecture Notes in Computer Science

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Notions of guardedness serve to delineate the admissibility of cycles, e.g. in recursion, corecursion, iteration, or tracing. We introduce an abstract notion of guardedness structure on a symmetric monoidal category, along with a corresponding notion of guarded traces, which are defined only if the cycles they induce are guarded. We relate structural guardedness, determined by propagating guardedness along the operations of the category, to geometric guardedness phrased in terms of a diagrammatic language. In our setup, the Cartesian case (recursion) and the co-Cartesian case (iteration) become completely dual, and we show that in these cases, guarded tracedness is equivalent to presence of a guarded Conway operator, in analogy to an observation on total traces by Hasegawa and Hyland. Moreover, we relate guarded traces to unguarded categorical uniform fixpoint operators in the style of Simpson and Plotkin. Finally, we show that partial traces based on Hilbert-Schmidt operators in the category of Hilbert spaces are an instance of guarded traces.

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Inversion, Iteration, and the Art of Dual Wielding

Kaarsgaard

2019

Reversible Computation

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Equational Properties of Fixed Point Operations in Cartesian Categories: An Overview

Cited by 7 publications

References 50 publications

Inversion, Iteration, and the Art of Dual Wielding

Inversion, Iteration, and the Art of Dual Wielding

Guarded Traced Categories

Inversion, Iteration, and the Art of Dual Wielding

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