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An Algebra of Dataflow Networks
 Fundamenta Informaticae
, 1995
"... . This paper describes an algebraic framework for the study of dataflow networks, which form a paradigm for concurrent computation in which a collection of concurrently and asynchronously executing processes communicate by sending messages between ports connected via FIFO message channels. A syntact ..."
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. This paper describes an algebraic framework for the study of dataflow networks, which form a paradigm for concurrent computation in which a collection of concurrently and asynchronously executing processes communicate by sending messages between ports connected via FIFO message channels. A syntactic dataflow calculus is defined, having two kinds of terms which represent networks and computations, respectively. By imposing suitable equivalences on networks and computations, we obtain the free dataflow algebra, in which the dataflow networks with m input ports and n output ports are regarded as the objects of a category S n m , and the computations of such networks are represented by the arrows. Functors defined on S n m label each computation by the input buffer consumed and the output buffer produced during that computation, so that each S n m is a span in Cat. It is shown that the free dataflow algebra construction underlies a monad in the category of collections S = fS n m : m...
Coherence for factorization algebras
 Theory Appl. Categories
, 2002
"... ABSTRACT. For the 2monad ((−) 2,I,C)onCAT, with unit I described by identities and multiplication C described by composition, we show that a functor F: K2 � � K satisfying FIK =1K admits a unique, normal, pseudoalgebra structure for (−) 2 if and only if there is a mere natural isomorphism F · F 2 ..."
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ABSTRACT. For the 2monad ((−) 2,I,C)onCAT, with unit I described by identities and multiplication C described by composition, we show that a functor F: K2 � � K satisfying FIK =1K admits a unique, normal, pseudoalgebra structure for (−) 2 if and only if there is a mere natural isomorphism F · F 2 � � � F · CK. We show that when this is the case the set of all natural transformations F · F 2 � � F · CK formsa commutative monoid isomorphic to the centre of K. 1. Preliminaries 1.1. When we speak of ‘the 2monad (−) 2 on CAT ’ we understand the canonical monad that arises by exponentiation of the cocommutative comonoid structure
THE COMPREHENSIVE FACTORIZATION AND TORSORS
, 2010
"... This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the tors ..."
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This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the torsors that H1 classifies are particular functors in E. We only impose conditions on E that are satisfied also by Cat(E) and argue that H1 for Cat(E) is a kind of H2 for E, and so on recursively. For us, it is too much to ask E to be a topos (or even internally complete) since, even if E is, Cat(E) is not. With this motivation, we are led to examine morphisms in E which act as internal families and to internalize the comprehensive factorization of functors into a nal functor followed by a discrete bration. We de ne Btorsors for a category B in E and prove clutching and classification theorems. The former theorem clutches ƒech cocycles to construct torsors while the latter constructs a coefficient category to classify structures locally isomorphic to members of a given internal family of structures. We conclude with applications to examples.
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
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Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.
Glueing Algebraic Structures on a 2Category
, 2000
"... We study the glueing constructions (comma objects) on general algebraic structures on a 2category, described in terms of 2monads and adjunctions. Specifically, lifting theorems for the comma objects and changeofbase results on both algebras of 2monads and adjunctions in a 2category are present ..."
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We study the glueing constructions (comma objects) on general algebraic structures on a 2category, described in terms of 2monads and adjunctions. Specifically, lifting theorems for the comma objects and changeofbase results on both algebras of 2monads and adjunctions in a 2category are presented. As a leading example, we take the 2monad on Cat whose algebras are symmetric monoidal categories, and show that many of the constructions in our previous work on models of linear type theories can be derived within this axiomatics. 1 Introduction In the previous work [2, 3] we have considered a glueing construction for symmetric monoidal (closed) categories, for studying the logical predicates for models of linear type theories. In that construction the glueing functor is supposed to be lax symmetric monoidal, thus preserves the structure only up to a few coherent morphisms, not up to isomorphisms or identity. From a view of the study of categories with algebraic structures [8] (which...
Functorial KripkeBethJoyal models of the lambda Picalculus I: type theory and internal logic
, 2001
"... We give a categorical account of KripkeBethJoyal models of the  calculus. Kripke models. Emphasize semantics of (terms/representatives/realizers for) consequences. 1 Introduction This paper, \Functorial KripkeBethJoyal models of the calculus I: type theory and internal logic" (henceforth abb ..."
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We give a categorical account of KripkeBethJoyal models of the  calculus. Kripke models. Emphasize semantics of (terms/representatives/realizers for) consequences. 1 Introduction This paper, \Functorial KripkeBethJoyal models of the calculus I: type theory and internal logic" (henceforth abbreviated to I), is rst of a sequence of three connected works. It is concerned with the basic model theory of the  calculus considered on the one hand as a system of rstorder dependent function types and on the other as presentation of the f8; gfragment of minimal rstorder predicate logic with proofobjects. From the point of view of type theory, ... MITCHELL/MOGGI From the point of view of logic, the ... At the core of our denition of Kripke(BethJoyal) models of lies our treatment of comprehension, context extension and (rstorder) dependent function spaces. The essential idea is similar that of earlier work [?, ?, ?]; however, our treatment has the following two advant...
Coherence For Factorization Algebras
, 2002
"... For the 2monad (() 2 , I, C) on CAT, with unit I described by identities and multiplication C described by composition, we show that a functor F : K 2 ## K satisfying F I K = 1 K admits a unique, normal, pseudoalgebra structure for () 2 if and only if there is a mere natural isomorp ..."
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For the 2monad (() 2 , I, C) on CAT, with unit I described by identities and multiplication C described by composition, we show that a functor F : K 2 ## K satisfying F I K = 1 K admits a unique, normal, pseudoalgebra structure for () 2 if and only if there is a mere natural isomorphism F F 2 # ## F CK . We show that when this is the case the set of all natural transformations F F 2 ## F CK forms a commutative monoid isomorphic to the centre of K.
Abstract. KAN EXTENSIONS IN DOUBLE CATEGORIES (ON WEAK DOUBLE CATEGORIES, PART III)
"... are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category. ..."
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are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category.
Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory
, 2002
"... In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction betwee ..."
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In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexedparacategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.