Revue de Synthèse

, Volume 124, Issue 1, pp 173–203 | Cite as

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

Articles

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 

List of references

  1. Abbott (Edwin A.), 1976,Flatland. A romance of many dimensions, 1st ed. London, 1884, here Cutchogue, NY, Buccaneer Books. The text is also available at <http://www.alcyone.com/max/lit/flatland/>.Google Scholar
  2. Al-Agl (Fahd),Brown (Ronald) andSteiner (Richard), 2002, «Multiple categories. The equivalence between a globular and cubical approach»,Advances in Mathematics, 170, p. 71–118.CrossRefGoogle Scholar
  3. Arzi-Gonczarowski (Zippora) andLehmann (Daniel), 1998, «The category of artificial perceptions»,Annals of Mathematics and Artificial Intelligence, 23, p. 267–298. (The site http://users.actcom.co.il/typographics/zippie/ contains other relevant 0129 0159 V work by the same authors.)CrossRefGoogle Scholar
  4. Atiyah (Michael), 2001, «Mathematics in the xxth century. Geometryversus Algebra»,Mathematics today, ser. 2,37, p. 47–49.Google Scholar
  5. Baez (John), 1995, «This week's finds in mathematical physics»,Week 53, 18 May (this text is also available at 〈http://math.ucr.edu/home/baez/twf.html〉).Google Scholar
  6. Baez (J.), 2001, «Higher-dimensional algebra and Planck-scale physics», inCallender (Craig) andHuggett (Nick), eds.,Physics meets philosophy at the Planck scale, Cambridge, Cambridge University Press, p. 177–195.Google Scholar
  7. Baez (J.), 2003*, 〈http://math.ucr.edu/home/baez/papers.html〉 (date of consultation).Google Scholar
  8. Brown (Ronald), 1967, «Groupoids and Van Kampen's theorem»,Proceedings of the London Mathematical Society, ser. 3,17, p. 385–401.CrossRefGoogle Scholar
  9. Brown (R.), 1968,Elements of modern topology, Maidstone, McGraw Hill.Google Scholar
  10. Brown (R.), 1982, «Higher dimensional group theory», inBrown (Ronald) andThickstun (Tom L.), eds.,Low dimensional topology, Cambridge, Cambridge University Press (London Mathematical Society Lecture Note Series, 48), p. 215–238.Google Scholar
  11. Brown (R.), 1987, «From groups to groupoids. A brief survey»,Bulletin London Mathematical Society, 19, p. 113–134.CrossRefGoogle Scholar
  12. Brown (R.), 1988,Topology. A geometric account of general topology, homotopy types and the fundamental groupoid, Chichester, Ellis Horwood.Google Scholar
  13. Brown (R.), 1992, «Out of line»,Royal Institution Proceedings, 64, p. 207–243.Google Scholar
  14. Brown (R.), 1996, «Higher dimensional group theory», web article available at 〈http://www.bangor.ac.uk/∼mas010/hdweb2.html〉Google Scholar
  15. Brown (Ronald),Hardie (Keith),Kamps (Heiner) andPorter (Timothy), 2002, «A homotopy double groupoid of a Hausdorff space»,Theory and Application of Categories, 10, p. 71–93.Google Scholar
  16. Brown (Ronald) andHiggins (Philip John), 1978, «On the connection between the second relative homotopy groups of some related spaces»,Proceedings of the London Mathematical Society, ser. 3,36, p. 193–212.CrossRefGoogle Scholar
  17. Brown (R.) andHiggins (P.J.), 1981a, «The algebra of cubes»,Journal of Pure and Applied Algebra, 21, p. 233–260.CrossRefGoogle Scholar
  18. Brown (R.) andHiggins (P.J.), 1981b, «Colimits of relative homotopy groups»,Journal of Pure and Applied Algebra, 22, p. 11–41.CrossRefGoogle Scholar
  19. Brown (Ronald) andLoday (Jean-Louis), 1987, «Van Kampen theorems for diagrams of spaces»,Topology, 26, p. 311–335.CrossRefGoogle Scholar
  20. Brown (Ronald) andMosa (Ghafar H.), 1999, «Double categories, 2-categories, thin structures and connections»,Theory and Applications of Categories, 5, p. 163–175.Google Scholar
  21. Brown (Ronald) andSpencer (Christopher B.), 1976, “Double groupoids and crossed modules»,Cathier de topologie et géométrie differentielle catégorique, 17, p. 343–362.Google Scholar
  22. Čech (Eduard), 1932, «Höherdimensionale homotopiegruppen»,Verhandlungen des Internationalen Mathematiker-Kongresses Zurich, Zurich-Leipzig, O. Füssli Verlag, vol. II, p. 203.Google Scholar
  23. Gonnes (Alain), 1994,Non-commutative geometry, transl. SterlingK. Berberian, San Diego (CA)-London, Academic Press.Google Scholar
  24. Cordier (Jean-Marc) andPorter (Timothy), 1989,Shape theory: Categorical methods of approximation, Chichester, Ellis Horwood (Mathematics and its Applications).Google Scholar
  25. Dobzhansky (Theodosius), 1973, «Nothing in biology makes sense except in the light of evolution»,The American Biology Teacher, 35, march, p. 125–129.Google Scholar
  26. Ehresmann (Charles), 1965,Catégories et structure, Paris, Dunod.Google Scholar
  27. Ehresmann (Andree) andVanbremeersch (Jean-Paul), 2001, «Emergence processes up to consciousness using the multiplicity principle and quantum physics»,American Institute of Physics Conference Proceedings, August. (See also other papers on <http://perso.wanadoo.fr/vbm-ehr/>.)Google Scholar
  28. Ferris (Timothy) andFadiman (Clifton), eds., 1991,The World Treasury of physics, astronomy and mathematics, Boston (MA)/London, Little/Brown and Co.Google Scholar
  29. Gadducci (Fabio) andMontanari (Ugo), 2000, «The tile model», inPlotkin (Gordon),Stirling (Colin) andTofte (Mads), eds.,Proof, language and interaction. Essays in honour of Robin Milner, Cambridge (MA)-London, MIT Press.Google Scholar
  30. Gaucher (Philippe), 2000, «Homotopy invariants of higher dimensional categories and concurrency in computer science»,Mathematical Structures in Computer Science, 10, p. 481–524.CrossRefGoogle Scholar
  31. Goubault (Éric), 2000, «Geometry and concurrency. A user's guide»,Mathematical Structures in Computer Science, 10, p. 411–425.CrossRefGoogle Scholar
  32. Hurewicz (Witold), 1935 and 1936, «Beiträge zur Topologie des Deformationen»,Nederlanden Akademie von Wetenschaften, ser. A,38, p. 112–119, 521–528, and39, p. 117–126, 213–224.Google Scholar
  33. Lakoff (George) andNúñez (Rafael), 2000,Where mathematics comes from. How the embodied mind brings mathematics into being, New York, NY, Basic Books.Google Scholar
  34. Lawvere (Friedrich William), 1992, «The space of mathematics. Philosophical, epistemological and historical explorations» inCategories of space and of quantily, International Symposium on structures in mathematical theories, San Sebastian, Spain, 1990, Berlin, De Gruyter, p. 14–30.Google Scholar
  35. Leinster (Tom), 2002, «A survey of definitions ofn-category»,Theory and Applications of Categories, 10, p. 1–70.Google Scholar
  36. Longo (Giuseppe), 1998, «Mathematics and the biological phenomena», inProceedings of the International Symposium on Foundations in Mathematics and Biology-Problems, prosects and interactions, invited lecture, Pontifical Lateran University, Vatican City, to appear.Google Scholar
  37. Longo (G.), 2001, «The reasonable effectiveness of mathematics and its cognitive roots», inBoi (Luciano), ed.,New interactions of mathematics with natural sciences and the humanities, Berlin-New York, Springer, to appear.Google Scholar
  38. Montanari (Ugo), 2003*, Publications web page, <http://www.di.unipi.it/ugo/ugo.html>.Google Scholar
  39. Petri (Carl Adam), 1973, «Concepts of net theory», inMathematical Foundations of Computer Science, Proceedings of Symposium and Summer School, Mathematical Institute of the Slovak Academy of Sciences, 3–8 Sept. 1973, High Tatras, Czechoslovakia, p. 137–146.Google Scholar
  40. Porter (Timothy), 1994, «Categorical shape theory as a formal language for pattern recognition»,Annals of Mathematics and Artificial Intelligence, 10, p. 25–54.CrossRefGoogle Scholar
  41. Pratt (Vaughan), 2003*, Guide to Papers on Chu Spaces, <http://chu.stanford.edu/guide.html>.Google Scholar
  42. Street (Ross), 1996, «Categorical structures» inHazewinkel (Michiel), ed.,Handbook of algebra, Amsterdam-Oxford, Elsevier, vol. I, p. 529–577.Google Scholar
  43. Wigner (Eugene Paul), 1991, «The unreasonable effectiveness of mathematics in the natural sciences», 1st ed. 1960,Communications in Pure and Applied Mathematics, 13, p. 1–14, here repr. inFerris andFadiman, 1991, p. 526–540.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wales, BangorBangorU.K.

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