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19
Tensor products and homotopies for ωgroupoids and crossed complexes
, 2007
"... Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed comp ..."
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Cited by 43 (21 self)
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Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves nonabelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ωgroupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.
Countable Lawvere Theories and Computational Effects
, 2006
"... Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere ..."
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Cited by 10 (2 self)
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Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.
Graphbased logic and sketches I: The general framework. Available by web browser from http://www.cwru.edu/1/class/mans/math/pub/wells
, 1996
"... Sketches as a method of specification of mathematical structures are an alternative to the stringbased specification employed in mathematical logic. ..."
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Cited by 8 (4 self)
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Sketches as a method of specification of mathematical structures are an alternative to the stringbased specification employed in mathematical logic.
Diagrammatic logic applied to a parameterization process
, 2009
"... Abstract. This paper provides an abstract definition of some kinds of logics, called diagrammatic logics, together with a definition of morphisms and of 2morphisms between diagrammatic logics. The definition of the 2category of diagrammatic logics rely on category theory, mainly on adjunction, cat ..."
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Cited by 4 (4 self)
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Abstract. This paper provides an abstract definition of some kinds of logics, called diagrammatic logics, together with a definition of morphisms and of 2morphisms between diagrammatic logics. The definition of the 2category of diagrammatic logics rely on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalization of a parameterization process. This process, which consists in adding a formal parameter to some operations in a given specification, is presented as a morphism of logics. Then the parameter passing process, for recovering a model of the given specification from a model of the parameterized specification and an actual parameter, is seen as a 2morphism of logics. 1
Sketches: Outline with References
 Dept. of Computer Science, Katholieke Universiteit Leuven
, 1994
"... This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions o ..."
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Cited by 2 (0 self)
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This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions of some of the less standard terms used in the paper, but the reader is expected to be familiar with the basic ideas of category theory. A rough machine generated index begins on page 21. I would have liked to explain the main ideas of all the papers referred to herein, but I am not familiar enough with some of them to do that. It seemed more useful to be inclusive, even if many papers were mentioned without comment. One consequence of this is that the discussions in this document often go into more detail about the papers published in North America than about those published elsewhere. The DVI file for this article is available by anonymous FTP from ftp.cwru.edu in the directory
An operadic approach to internal structures
"... We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where ..."
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Cited by 2 (2 self)
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We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where
Diagrammatic Inference
, 710
"... Abstract. Diagrammatic logics were introduced in 2002, with emphasis on the notions of specifications and models. In this paper we improve the description of the inference process, which is seen as a Yoneda functor on a bicategory of fractions. A diagrammatic logic is defined from a morphism of limi ..."
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Cited by 1 (1 self)
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Abstract. Diagrammatic logics were introduced in 2002, with emphasis on the notions of specifications and models. In this paper we improve the description of the inference process, which is seen as a Yoneda functor on a bicategory of fractions. A diagrammatic logic is defined from a morphism of limit sketches (called a propagator) which gives rise to an adjunction, which in turn determines a bicategory of fractions. The propagator, the adjunction and the bicategory provide respectively the syntax, the models and the inference process for the logic. Then diagrammatic logics and their morphisms are applied to the semantics of side effects in computer languages.
Modular Specifications: Constructions With Finite Colimits, Diagrams, Isomorphisms
, 1996
"... : The composition of modular specifications can be modeled, in a category theoretic framework, by colimits of diagrams. Pushouts in particular describe the combination of two specifications sharing a common part. This work extends this classic idea along three lines. First, we define a term language ..."
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: The composition of modular specifications can be modeled, in a category theoretic framework, by colimits of diagrams. Pushouts in particular describe the combination of two specifications sharing a common part. This work extends this classic idea along three lines. First, we define a term language to represent modular specifications built with colimit constructions over a category of base specifications. This language is formally characterized by a finitely cocomplete category. Then, we propose to associate with each term a diagram. This interpretation provides us with a more abstract representation of modular specifications because irrelevant steps of the construction are eliminated. We define a category of diagrams, which is a completion of the base category with finite colimits. We prove that the interpretation of terms as diagrams defines an equivalence between the corresponding categories, which shows the correctness of this interpretation. At last, we propose an algorithm to no...
DISTRIBUTIVE LAWS IN PROGRAMMING STRUCTURES
, 2009
"... Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approac ..."
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Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2naturality and Gray’s tensor product of 2categories. It generalises the existing more specific notions of distributive