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The intuitions of higher dimensional algebra for the study of structured space

Ronald Brown¹ and Timothy Porter¹

(1)	Department of Mathematics, University of Wales, Bangor, Dean Street, LL57 1 UT Bangor, Gwynedd, U.K.

Abstract Higher dimensional algebra frees mathematics from the restriction to a purely linear notation, in order to improve the modelling of geometry and so obtain more understanding and more modes of computation. It gives new tools for noncommutative, higher dimensional, local to global problems, through the notion of «algebraic inverse to subdivision». We explain the way these ideas arose for the writters, in extending first the classical notion of abstract group to abstract groupoid, in which composition is only partially defined, as in composing journeys, and which brings a spatial component to the usual group theory: An example from knot theory is used to explain how such algebra can be used to describe some structure of a space. The extension to dimension 2 uses compositions of squares in two directions, and the richness of the resulting algebra is shown by some 2-dimensional calculations. The difficulty of the jump from dimension 1 to dimension 2 is also illustrated by the comparison of the commutative square with the commutative cube-discussion of the latter requires new ideas. The importance of category theory is explained, and a range of current and potential applications of higher dimensional algebra indicated.

Keywords higher dimensional algebra - knot theory - groups - groupoids - category theory - connections - cubical methods

Résumé Les algèbres de dimensions supérieures libèrent les mathématiques de la restriction d'une notation purement linéaire. Elles aident ainsi à la modélisation de la géométrie et procurent une meilleure compréhension et plus de possibilitiés pour les calculs. Elles nous donnent de nouveaux outils pour l;étude de problèmes non-commotatifs., de dimension supérieure que assurent le passage du local au global, en utilisant la notion d' «inverse algébrique de subdivision». Nous allons exposer comment ces idées sont venues aux auteurs en prolongeant initialement la notion classique de groupe abstrait à celle de groupoïde abstrait, dont la composition n'est que partiellement définie, et qui ajoute une composante spatiale à la théorie habituelle des groupes. La théorie des noeuds nous fournit un exemple en indiquant comment une telle algèbre peut être utilisée pour décrire la structure d'un espace. Le prolongement à la dimension 2 utilise des compositions de carrés dans deux directions et la richesse de l'algèbre qui en résulte est montrée par certains calculs de dimension 2. La difficulté de la transition de la dimension 1 à la dimension 2 est également illustrée par la comparaison de la notion de carré commutatif à celle de cube commutatif — le traitement de cette derniére nécessitant de nouvelles notions. L'importance de la théorie des caté gories est expliquée, de même que les possibilités de l'application d'algèbres de dimensions supérieures.

Mots-Clés algébres de dimensions supérieures - théorie des nœuds - groupes - groupoïdes - théorie des catégories - connections - méthodes cubiques

Zusammenfassung Die mehrdimensionale Algebra befreit die Mathematik von einer rein linearen Notation. Sie ermöglicht eine Modellierung der Geometrie, erleichtert das Verständnis von Rechnungen und stellt dafür eine größere Zahl von Hilfsmitteln zur Verfügung. Sie liefert uns auch neue Werkzeuge für die Untersuchung «lokal-globaler» nichtkommutativer Probleme, indem der Begriff einer algebraischen Umkehrung der Subdivision eingeführt wird. Wir werden zeigen, wie diese Vorstellungen entstanden sind. Der klassische, abstrakte Grupenbegriff wurde zu einem abstrakten «Gruppoid» weiterentwickelt, dessen Zusammensetzung nur teilweise definiert ist und bei dem der herkömmlichen Gruppentheorie eine räumliche Komponente hinzugefügt wird. Die Knotentheorie liefert dafür ein Beispiel, denn sie zeigt den Nutzen einer solchen Algebra für die Beschreibung einer Raumstruktur. Bei der Ausweitung auf die zweite Dimension werden in zwei Richtungen Zusammensetzungen von Quadraten benutzt, und der Reichtum der daraus hervorgehenden Algebra wird durch einige Rechnungen in der zweiten Dimension erläutert. Die Schwierigkeit des Übergangs von der ersten zur zweiten Dimension wird durch den Vergleich des Begriffs «kommutatives Quadrat» mit dem Begriff «kommutativer Würfel» erläutert. Um diesen zu behandeln, sind neue Ideen erforderlich. Ferner wird die Bedeutung der Theorie der Kategorien erklärt, und es wird eine Reihe von bereits existierenden und potentiellen Anwendungen der mehrdimensionalen Algebra angegeben.

Stichwörter mehrdimensionale Algebra - Knotentheorie - Gruppe - Gruppoide - Theorie der Kategorien - Verknüpfungen - kubische Methoden

Riassunto L'algebra di dimensioni superiori libera la matematica dai limiti delle notazioni puramente lineari, al fine di migliorare la modellizzazione della geometria ed ottenere una migliore comprensione e migliori strumenti di calcolo. Fornisce inoltre nuovi strumenti per strutture non-commutative, dimensioni superiori e passaggi dal locale al globale, grazie alla nozione di «inverso algebrico della suddivisione». Spieghiamo il modo in cui gli autori sono pervenuti a queste idee, estendendo in primo luogo la nozione classica di gruppo astratto a quella di gruppoide astratto, nel quale la composizione è definita solo parzialmente, come nei «percorsi di composizione» in cui si aggiunge una componenente spaziale alla usuale teoria dei gruppi. Viene usato un esempio della teoria dei nodi al fine di spiegare come una tale algebra possa essere usata al fine di descrivere alcune strutture spaziali. L'estensione a due dimensioni usa la composizione dei quadrati in due direzioni e la ricchezza dell'algebra corrispondente è dimostrata da alcuni conti in due dimensioni. La difficoltà del passare da una a due dimensioni è anche illustrata dal raffronto del quadrato commutativo con il cubo commutativo, la cui discussione richiede concetti nuovi. Si spiega infine l'importanza della teoria delle categorie e lo spazio delle applicazioni attuali e possibili dell'algebra di dimensioni superiori.

Parole Chiave algebra di dimensioni superiori - teoria dei nodi - gruppi - gruppoidi - teoria delle categorie - connessione - metodi cubici

RonaldBrown, born in 1935, is emeritus professor, «Leverhulme Emeritus Fellow» in the Department of Mathematics, at the University of Wales, Bangor. His research concerns topology, the theory of categories and their application.

TimothyPorter, born in 1947, is professor of mathematics in the Department of Mathematics, University of Wales Bangor. His research concerns topology, logic, the theory of categories and their application.

This is a development of a talk by Ronald Brown at the École normale supérieure on 30 May 2001, in the seminar series «Géométrie et Cognition» of Giuseppe Longo, Jean Petitot, Bernard Teissier.

We would like to thank: Giuseppe Longo for the invitation to lecture to his seminar; Posina Venkata Rayudu of the National Brain Research Centre, Gurgaon, India, for helpful comments; and Andree Ehresmann for helpful comments and additional references —readers interested in applications of category theory in biology should explore the web site perso.wanadoo.fr/vbm-ehr. The works of Charles Ehresmann in category theory and of John Henry Constantine White head in topology have been a strong influence in the development of this theory. We are grateful also to Laurie Catteeuw for her help in preparing the paper for this publication.

Ronald Brown (Corresponding author)
Email: r.brown@bangor.ac.uk

Timothy Porter
Email: t.porter@bangor.ac.uk

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