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Mathematical models of computational and combinatorial structures. Invited address for Foundations
 of Software Science and Computation Structures (FOSSACS 2005
, 2005
"... Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category ..."
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Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category theory, domain theory, logic, type theory, etc. In support of this proposal I will show how such an approach leads to interesting connections between various areas of computer science and mathematics; concentrating on one such example in some detail. Specifically, I will consider the line of my research involving denotational models of the pi calculus and algebraic theories with variablebinding operators, indicating how the abstract mathematical structure underlying these models fits with that of Joyal’s combinatorial species of structures. This analysis suggests both the unification and generalisation of models, and in the latter vein I will introduce generalised species of structures and their calculus. These generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc.) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.
ALGEBRAIC CATEGORIES WHOSE PROJECTIVES ARE EXPLICITLY FREE
"... Abstract. Let M = (M, m, u) be a monad and let (MX, m) be the free Malgebra on the object X. Consider an Malgebra (A, a), a retraction r: (MX, m) → (A, a) and a section t: (A, a) → (MX, m) of r. The retract (A, a) is not free in general. We observe that for many monads with a ‘combinatorial flav ..."
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Abstract. Let M = (M, m, u) be a monad and let (MX, m) be the free Malgebra on the object X. Consider an Malgebra (A, a), a retraction r: (MX, m) → (A, a) and a section t: (A, a) → (MX, m) of r. The retract (A, a) is not free in general. We observe that for many monads with a ‘combinatorial flavor ’ such a retract is not only a free algebra (MA0, m), but it is also the case that the object A0 of generators is determined in a canonical way by the section t. We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory. 1.
REFLECTIVE KLEISLI SUBCATEGORIES OF THE CATEGORY OF EILENBERGMOORE ALGEBRAS FOR FACTORIZATION MONADS Contents
"... ABSTRACT. It is well known that for any monad, the associated Kleisli category is embedded in the category of EilenbergMoore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left ..."
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ABSTRACT. It is well known that for any monad, the associated Kleisli category is embedded in the category of EilenbergMoore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left
DECOMPOSABLE FUNCTORS AND THE EXPONENTIAL PRINCIPLE, II
"... We develop a new setting for the exponential principle in the context of multisort species, where indecomposable objects are generated intrinsically instead of being given in advance. Our approach uses the language of functors and natural transformations (composition operators), and we show that, ..."
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We develop a new setting for the exponential principle in the context of multisort species, where indecomposable objects are generated intrinsically instead of being given in advance. Our approach uses the language of functors and natural transformations (composition operators), and we show that, somewhat surprisingly, a single axiom for the composition already suffices to guarantee validity of the exponential formula. We provide various illustrations of our theory, among which are applications to the enumeration of (semi)magic squares.