Categorical semantics for arrows

Journal of Functional Programming (Impact Factor: 0.52). 07/2009; 19(3-4):403-438. DOI: 10.1017/S0956796809007308
Source: DBLP


Arrows are an extension of the well-established notion of a monad in functional programming languages. This article presents several examples and constructions, and develops
denotational semantics of arrows as monoids in categories of bifunctors C^op x C -> C. Observing similarities to monads -- which are monoids in categories of endofunctors C -> C
-- it then considers Eilenberg-Moore and Kleisli constructions for arrows. The latter yields
Freyd categories, mathematically formulating the folklore claim “arrows are Freyd categories”.

Download full-text


Available from: Chris Heunen, Sep 30, 2015
24 Reads
  • Source
    • "The representation theorem does not really use the inverse operation of groups so one can generalise the representation to monoids, yielding a Cayley representation theorem for monoids (Jacobson, 2009). "
    [Show abstract] [Hide abstract]
    ABSTRACT: There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article we show that these three notions can be seen as monoids in a monoidal category. We demonstrate that at this level of abstraction one can obtain useful results which can be instantiated to the different notions of computation. In particular, we show how free constructions and Cayley representations for monoids translate into useful constructions for monads, applicative functors, and arrows. Moreover, the uniform presentation of all three notions helps in the analysis of the relation between them.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This paper contributes to the theory of coalgebraic, state-based modelling of components via two additions: a feedback operator in the form of a monoidal trace, and a three-dimensional string calculus for representing and manipulating composite component diagrams. The feedback operator on components is shown to satisfy the trace axioms by Joyal, Street and Verity. As a corollary, we appeal to the microcosm prin-ciple and derive a canonical traced monoidal structure on the category of resumptions. This generalises an observation by Abramsky, Haghverdi and Scott.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior sys- tematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalge- braic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic struc- ture on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional pro- gramming.
    Algebra and Coalgebra in Computer Science, Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009. Proceedings; 09/2009
Show more