Applied Categorical Structures – Official Blog

by Davey, B. A.; Haviar, M.; Priestley, H. A.

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices.

DOI: 10.1007/s10485-007-9090-7
Online Date: 5/24/2007
Print publication date: 6/1/2007
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by Grandis, Marco

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the modelling of biological systems have emerged.

DOI: 10.1007/s10485-007-9084-5
Online Date: 5/23/2007
Print publication date: 8/1/2007
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by Lafont, Yves

We present various results of the last 20 years converging towards a homotopical theory of computation. This new theory is based on two crucial notions: polygraphs (introduced by Albert Burroni) and polygraphic resolutions (introduced by François Métayer). There are two motivations for such a theory:
Providing invariants of computational systems to study those systems and prove properties about them;Finding new methods to make computations in algebraic structures coming from geometry or topology.This means that this theory should be relevant for mathematicians as well as for theoretical computer scientists, since both may find useful tools or concepts for their own domain coming from the other one.

DOI: 10.1007/s10485-007-9083-6
Online Date: 5/4/2007
Print publication date: 8/1/2007
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by Beggs, Edwin J.; Brzeziński, Tomasz

An explicit formula for a strong connection form in a principal extension by a coseparable coalgebra is given.

DOI: 10.1007/s10485-007-9087-2
Online Date: 5/3/2007
Print publication date: 4/1/2008
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by Wisbauer, Robert

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of (
F
,
G
)-dimodules associated to two functors $F,G:\mathbb{A}\to \mathbb{B}$ between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.

DOI: 10.1007/s10485-007-9076-5
Online Date: 5/3/2007
Print publication date: 4/1/2008
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by Weber, Mark

A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the Yoneda structure (Street and Walters, J. Algebra, 50:350–379, 1978) it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a Yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category (Street, Lecture Notes in Math., 420:104–133, 1974) and provides a self-contained development of the necessary background material on Yoneda structures.

DOI: 10.1007/s10485-007-9079-2
Online Date: 5/3/2007
Print publication date: 6/1/2007
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