7 research outputs found

    Central Submonads and Notions of Computation

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    The notion of "centre" has been introduced for many algebraic structures in mathematics. A notable example is the centre of a monoid which always determines a commutative submonoid. Monads in category theory are important algebraic structures that may be used to model computational effects in programming languages and in this paper we show how the notion of centre may be extended to strong monads acting on symmetric monoidal categories. We show that the centre of a strong monad T\mathcal T, if it exists, determines a commutative submonad Z\mathcal Z of T\mathcal T, such that the Kleisli category of Z\mathcal Z is isomorphic to the premonoidal centre (in the sense of Power and Robinson) of the Kleisli category of T\mathcal T. We provide three equivalent conditions which characterise the existence of the centre of T\mathcal T and we show that every strong monad on many well-known naturally occurring categories does admit a centre, thereby showing that this new notion is ubiquitous. We also provide a computational interpretation of our ideas which consists in giving a refinement of Moggi's monadic metalanguage. The added benefit is that this allows us to immediately establish a large class of contextually equivalent programs for computational effects that are described via monads that admit a non-trivial centre by simply considering the richer syntactic structure provided by the refinement.Comment: 25 pages + 8 pages references and appendi

    Diagrammatic presentations of enriched monads and varieties for a subcategory of arities

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    The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke-Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a VV-category CC in terms of parametrized JJ-ary operations and diagrammatic equations for a suitable subcategory of arities J↪CJ \hookrightarrow C. On this basis we introduce the notions of diagrammatic JJ-presentation and JJ-ary variety, and we show that the category of JJ-ary varieties is dually equivalent to the category of JJ-ary VV-monads. We establish several examples of diagrammatic JJ-presentations and JJ-ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic JJ-presentations. We show that both JJ-relative monads and JJ-pretheories give rise to diagrammatic JJ-presentations that directly describe their algebras. Using diagrammatic JJ-presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff's Galois connection between classes of algebras and sets of equations to the above setting

    Free algebras of topologically enriched multi-sorted equational theories

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    Classical multi-sorted equational theories and their free algebras have been fundamental in mathematics and computer science. In this paper, we present a generalization of multi-sorted equational theories from the classical (SetSet-enriched) context to the context of enrichment in a symmetric monoidal category VV that is topological over SetSet. Prominent examples of such categories include: various categories of topological and measurable spaces; the categories of models of relational Horn theories without equality, including the categories of preordered sets and (extended) pseudo-metric spaces; and the categories of quasispaces (a.k.a. concrete sheaves) on concrete sites, which have recently attracted interest in the study of programming language semantics. Given such a category VV, we define a notion of VV-enriched multi-sorted equational theory. We show that every VV-enriched multi-sorted equational theory TT has an underlying classical multi-sorted equational theory ∣T∣|T|, and that free TT-algebras may be obtained as suitable liftings of free ∣T∣|T|-algebras. We establish explicit and concrete descriptions of free TT-algebras, which have a convenient inductive character when VV is cartesian closed. We provide several examples of VV-enriched multi-sorted equational theories, and we also discuss the close connection between these theories and the presentations of VV-enriched algebraic theories and monads studied in recent papers by the author and Lucyshyn-Wright.Comment: 51 pages plus six page Appendix. Revised to include more discussion of enrichment of categories of algebras (expanded Remarks 3.1.4 and 4.1.6; added more details to Section 6; expanded item 6.9; added an extra item to Theorem 6.10; shortened Remark 6.11
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