Central Submonads and Notions of Computation

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 $\mathcal T$, if it exists, determines a commutative submonad $\mathcal Z$ of $\mathcal T$, such that the Kleisli category of $\mathcal Z$ is isomorphic to the premonoidal centre (in the sense of Power and Robinson) of the Kleisli category of $\mathcal T$. We provide three equivalent conditions which characterise the existence of the centre of $\mathcal T$ 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

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 $V$-category $C$ in terms of parametrized $J$-ary operations and diagrammatic equations for a suitable subcategory of arities $J \hookrightarrow C$. On this basis we introduce the notions of diagrammatic $J$-presentation and $J$-ary variety, and we show that the category of $J$-ary varieties is dually equivalent to the category of $J$-ary $V$-monads. We establish several examples of diagrammatic $J$-presentations and $J$-ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic $J$-presentations. We show that both $J$-relative monads and $J$-pretheories give rise to diagrammatic $J$-presentations that directly describe their algebras. Using diagrammatic $J$-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

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 ($Set$-enriched) context to the context of enrichment in a symmetric monoidal category $V$ that is topological over $Set$. 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 $V$, we define a notion of $V$-enriched multi-sorted equational theory. We show that every $V$-enriched multi-sorted equational theory $T$ has an underlying classical multi-sorted equational theory $|T|$, and that free $T$-algebras may be obtained as suitable liftings of free $|T|$-algebras. We establish explicit and concrete descriptions of free $T$-algebras, which have a convenient inductive character when $V$ is cartesian closed. We provide several examples of $V$-enriched multi-sorted equational theories, and we also discuss the close connection between these theories and the presentations of $V$-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