5,040 research outputs found

    Variations on Algebra: monadicity and generalisations of equational theories

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    Dedicated to Rod Burstal

    Proto-exact categories of matroids, Hall algebras, and K-theory

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    This paper examines the category Mat\mathbf{Mat}_{\bullet} of pointed matroids and strong maps from the point of view of Hall algebras. We show that Mat\mathbf{Mat}_{\bullet} has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory K(Mat)K_* (\mathbf{Mat}_{\bullet}) of Mat\mathbf{Mat}_{\bullet} via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections πns(S)Kn(Mat)\pi^s_n (\mathbb{S}) \hookrightarrow K_n (\mathbf{Mat}_{\bullet}) from the stable homotopy groups of spheres for all nn. Finally, we show that the Hall algebra of Mat\mathbf{Mat}_{\bullet} is a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.Comment: 29 page

    Towards a Convenient Category of Topological Domains

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    We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

    A Convenient Category of Domains

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    We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual omegaomega-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

    Cartesian closed 2-categories and permutation equivalence in higher-order rewriting

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    We propose a semantics for permutation equivalence in higher-order rewriting. This semantics takes place in cartesian closed 2-categories, and is proved sound and complete
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