Cartesian effect categories are Freyd-categories

Authors:
Jean-Guillaume Dumas;Dominique Duval;Jean-Claude Reynaud
Affiliations:
LJK, Université de Grenoble, 38041 Grenoble, France;LJK, Université de Grenoble, 38041 Grenoble, France;Malhivert, 38640 Claix, France
Venue:
Journal of Symbolic Computation
Year:
2011

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Abstract

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.