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Arrows, like monads, are monoids
 Proc. of 22nd Ann. Conf. on Mathematical Foundations of Programming Semantics, MFPS XXII, v. 158 of Electron. Notes in Theoret. Comput. Sci
, 2006
"... Monads are by now wellestablished as programming construct in functional languages. Recently, the notion of “Arrow ” was introduced by Hughes as an extension, not with one, but with two type parameters. At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fai ..."
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Cited by 12 (1 self)
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Monads are by now wellestablished as programming construct in functional languages. Recently, the notion of “Arrow ” was introduced by Hughes as an extension, not with one, but with two type parameters. At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fairly civilised, by showing that they correspond to monoids in suitable subcategories of bifunctors C op ×C → C. This shows that, at a suitable level of abstraction, arrows are like monads — which are monoids in categories of functors C → C. Freyd categories have been introduced by Power and Robinson to model computational effects, well before Hughes ’ Arrows appeared. It is often claimed (informally) that Arrows are simply Freyd categories. We shall make this claim precise by showing how monoids in categories of bifunctors exactly correspond to Freyd categories.
Traced Premonoidal Categories
, 1999
"... Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating trace ..."
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Cited by 7 (0 self)
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Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating traces and Conway operators in cartesian categories.
Cartesian effect categories are Freydcategories
, 2009
"... 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 sideeffects. In this paper Cartesian e ..."
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Cited by 6 (6 self)
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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 sideeffects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freydcategories and Haskell’s Arrows. It is proved that a Cartesian effect category is a Freydcategory 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. Keywords. Categorical logic, computational effects, monads, Freydcategories, premonoidal categories, Arrows, sequential product, effect categories, Cartesian effect categories.
A semantical approach to equilibria and rationality
, 905
"... ”An equilibrium does not appear because agents are rational, but rather agents appear to be rational because an equilibrium has been reached.[...] The task for game theory is to formulate a notion of rationality.” Larry Samuelson [20, p. 3] Abstract. Game theoretic equilibria are mathematical expres ..."
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Cited by 1 (1 self)
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”An equilibrium does not appear because agents are rational, but rather agents appear to be rational because an equilibrium has been reached.[...] The task for game theory is to formulate a notion of rationality.” Larry Samuelson [20, p. 3] Abstract. Game theoretic equilibria are mathematical expressions of rationality. Rational agents are used to model not only humans and their software representatives, but also organisms, populations, species and genes, interacting with each other and with the environment. Rational behaviors are achieved not only through conscious reasoning, but also through spontaneous stabilization at equilibrium points. Formal theories of rationality are usually guided by informal intuitions, which are acquired by observing some concrete economic, biological, or network processes. Treating such processes as instances of computation, we reconstruct and refine some basic notions of equilibrium and rationality from the some basic structures of computation. It is, of course, well known that equilibria arise as fixed points; the point is that semantics of computation of fixed points seems to be providing novel methods, algebraic and coalgebraic, for reasoning about them. 1