Skip Navigation

International Mathematics Research Papers (2006) Vol. 2006 : article ID 69163, 54 pages, doi:10.1155/IMRP/2006/69163
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Kock, J.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Weak identity arrows in higher categories

Joachim Kock

Departament de Matemàtiques, Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona), Spain E-mail address: kock{at}mat.uab.es

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where, instead, the notion of identity arrow is weakened—these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category {Delta} is replaced by a certain "fat" delta of "coloured ordinals," where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad mathematical readership and contains also a brief introduction to simplicial viewpoints on higher categories in general. It is explained how the definition of fair n-category is almost forced upon us by three standard ideas. The second part states some basic results about fair categories, and give examples, including Moore path spaces and cobordism categories. The category of fair 2-categories is shown to be equivalent to the category of bicategories with strict composition laws. Fair 3-categories correspond to tricategories with strict composition laws. The main motivation for the theory is Simpson's weak-unit conjecture according to which n-groupoids with strict composition laws and weak units should model all homotopy n-types. A proof of a version of this conjecture in dimension 3 is announced, obtained in joint work with A. Joyal. Technical details and a fuller treatment of the applications will appear elsewhere.



References

  1. Artin M., Grothendieck A., Verdier J. L., Deligne P., Saint-Donat B., eds. Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. In: Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Lecture Notes in Mathematics (1972) 269. Berlin: Springer. xix+525.
  2. Baez J. C., Dolan J. Higher-dimensional algebra and topological quantum field theory. Journal of Mathematical Physics (1995) 36(11):6073–6105. Baez{at}math.ucr.edu, jdolan{at}math.ucr.edu.[CrossRef][Web of Science]
  3. Bénabou J. Introduction to bicategories. In: Reports of the Midwest Category Seminar (1967) Berlin: Springer. 1–77. jean.benabou{at}wanadoo.fr.
  4. Brown R., Higgins P. J. The equivalence of {infty}-groupoids and crossed complexes. Cahiers de Topologie et Géométrie Différentielle (1981) 22(4):371–386. r.brown{at}bangor.ac.uk, p.j.higgins{at}durham.ac.uk.
  5. Dwyer W. G., Kan D. M. Simplicial localizations of categories. Journal of Pure and Applied Algebra (1980) 17(3):267–284. dwyer.1{at}nd.edu.[CrossRef][Web of Science]
  6. Dwyer W. G., Kan D. M., Smith J. H. Homotopy commutative diagrams and their realizations. Journal of Pure and Applied Algebra (1989) 57(1):5–24. jhs{at}math.purdue.edu.[CrossRef][Web of Science]
  7. Elmendorf A. D., Kríz I., Mandell M. A., May J. P. Rings, Modules, and Algebras in Stable Homotopy Theory. In: Mathematical Surveys and Monographs (1997) 47. Rhode Island: American Mathematical Society. xii+249. with an appendix by M. Cole aelmendo{at}calumet.purdue.edu, ikriz{at}math.lsa.umich.edu, mandell{at}math.uchicago.edu, may{at}math.uchicago.edu.
  8. Gabriel P., Zisman M. Calculus of Fractions and Homotopy Theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (1967) 35. New York: Springer. x+168. gabriel{at}math.unizh.ch, zisman{at}math.jussieu.fr.
  9. Gordon R., Power A. J., Street R. Coherence for tricategories. Memoirs of the American Mathematical Society (1995) 117(558):vi+81. ajp{at}inf.ed.ac.uk, street{at}maths.mq.edu.au.
  10. Grothendieck A. Technique de descente et théorèmes d'existence en géometrie algébrique. III. Préschémas quotients. Séminaire Bourbaki, Exp. No. 212 (1960/61).
  11. Grothendieck A. Pursuing stacks. Letter to D. Quillen, 1983, 600pp.
  12. Hirschowitz A., Simpson C. Descente pour les n-champs (Descent for n-stacks). preprint, 2001, http://arxiv.org/abs/math.AG/9807049 ah{at}math.unice.fr.
  13. Hovey M. Model Categories. In: Mathematical Surveys and Monographs (1999) 63. Rhode Island: American Mathematical Society. xii+209. mhovey{at}wesleyan.edu.
  14. Joyal A., Kock J. Coherence for weak units. in preparation.
  15. Joyal A., Kock J. Weak units and homotopy 3-types. preprint, 2006, http://arxiv.org/abs/math.CT/0602084 joyal{at}math.uqam.ca.
  16. Kapranov M. M., Voevodsky V. A. {infty}-groupoids and homotopy types. Cahiers de Topologie et Géométrie Différentielle Catégoriques (1991) 32(1):29–46. kapranov{at}math.toronto.edu, vladimir{at}math.ias.edu.
  17. Kock J. Frobenius Algebras and 2D Topological Quantum Field Theories. In: London Mathematical Society Student Texts, no. 59 (2003) Cambridge: Cambridge University Press. xiv+240.
  18. Kock J. Categories of coloured ordinals. in preparation.
  19. Kock J. Commutativity in double semi-groups. preprint, 2004.
  20. Kock J. Elementary remarks on units in monoidal categories. preprint, 2005, http://arxiv.org/abs/math.CT/0507349.
  21. Kock J., Toën B. Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture. Compositio Mathematica (2005) 141(1):253–261.[CrossRef][Web of Science]
  22. Kríz I., May J. P. Operads, algebras, modules and motives. Astérisque (1995) (233):iv+145.
  23. Lawrence R. J. Triangulations, categories and extended topological field theories. In: Quantum Topology (1993) 3. New Jersey: World Scientific. 191–208. Ser. Knots Everything. ruthel{at}math.huji.ac.il.
  24. Leinster T. A survey of definitions of n-category. Theory and Applications of Categories (2002) 10(1):1–70.
  25. Leinster T. Higher Operads, Higher Categories. In: London Mathematical Society Lecture Note Series (2004) 298. Cambridge: Cambridge University Press. xiv+433. leinster{at}dpmms.cam.ac.uk.
  26. Mac Lane S. Categories for the Working Mathematician. In: Graduate Texts in Mathematics (1998) 5, 2nd. New York: Springer. xii+314.
  27. Pellissier R. Weak enriched categories—Categories enrichies faibles (2002) Université de Nice Sophia-Antipolis. http://arxiv.org/abs/math.AT/0308246 rpelliss{at}math.unice.fr.
  28. Saavedra Rivano N. Catégories Tannakiennes. In: Lecture Notes in Mathematics (1972) 265. Berlin: Springer. ii+418. neantro{at}sk.tsukuba.ac.jp.
  29. Segal G. Categories and cohomology theories. Topology. An International Journal of Mathematics (1974) 13(3):293–312. segalg{at}maths.ox.ac.uk.
  30. Simpson C. A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van Kampen. preprint, 2000, http://arxiv.org/abs/alg-geom/9704006.
  31. Simpson C. Homotopy types of strict 3-groupoids. preprint, 1998, http://arxiv.org/abs/math.CT/9810059 carlos{at}math.unice.fr.
  32. Simpson C. On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak n-categories. preprint, 1998, http://arxiv.org/abs/math.CT/9810058.
  33. Spitzweck M. Operads, algebras and modules in general model categories. preprint, 2001, 48 pages, part of Ph.D. thesis, http://arxiv.org/abs/math.AT/0101102 spitz{at}uni-math.gwdg.de.
  34. Tamsamani Z. Sur des notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux. K-Theory (1999) 16(1):51–99. ztamsama{at}math.uci.edu.
  35. Toën B. Vers une axiomatisation de la théorie des catégories supérieures. K-Theory (2005) 34(3):233–263. toen{at}picard.ups-tlse.fr.
  36. Toën B., Vezzosi G. Homotopical algebraic geometry. I. Topos theory. Advances in Mathematics (2005) 193(2):257–372. vezzosi{at}dma.unifi.it.[CrossRef][Web of Science]
  37. Toën B., Vezzosi G. Segal topoi and stacks over Segal categories. to appear in proceedings of the program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI Berkeley, January–May 2002, http://arxiv.org/abs/math.AG/0212330.

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Kock, J.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?