Cartesian effect categories are Freyd-categories

Jean-Guillaume Dumas (1) , Dominique Duval (1) , Jean-Claude Reynaud (2)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques
2 RC - Reynaud Consulting

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskell's Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions.

Mots clés: categorical logic; computational effects; monads; Freyd-categories; premonoidal categories; Arrows; sequential product; effect categories; Cartesian effect categories

Domaines: Logique en informatique [cs.LO]; Catégories et ensembles [math.CT]

Jean-Guillaume Dumas, Dominique Duval, Jean-Claude Reynaud. Cartesian effect categories are Freyd-categories. Journal of Symbolic Computation, 2011, 46 (3), pp.272-293. ⟨10.1016/j.jsc.2010.09.008⟩. ⟨hal-00369328v3⟩