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Applied Categorical Structures

, Volume 15, Issue 4, pp 439–453 | Cite as

An ω-category with all Duals is an ω-groupoid

  • Eugenia Cheng
Article

Abstract

We make a definition of ω-precategory which should underlie any definition of weak ω-category. We make a precise definition of pseudo-invertible cells in this setting. We show that in an ω-precategory with all weak duals, every cell is pseudo-invertible. We deduce that in any “sensible” theory of ω-categories, an ω-category with all weak duals is an ω-groupoid. We discuss various examples and open questions involving higher-dimensional tangles and cobordisms.

Keywords

Duals ω-category ω-groupoid ω-precategory 

Mathematics Subject Classifications (2000)

18A05 18D05 18D10 18D50 57R90 

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References

  1. 1.
    Baez, J., Dolan, J.: Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36, 6073–6105 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baez, J., Dolan, J.: Higher-dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math. 135(2), 145–206 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baez, J., Langford, L.: Higher-dimensional algebra IV: 2-tangles. Adv. Math. 180, 705–764 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cheng, E., Gurski, N.: Towards an n-category of cobordisms. In: CT06, White Point, June (2006)Google Scholar
  5. 5.
    Penon, J.: Approche polygraphique des ∞-catégories non strictes. Cahiers Topologie Géom. Différentielle Catég. XL(1), 31–80 (1999)MathSciNetGoogle Scholar
  6. 6.
    Shum, M.C.: Tortile tensor categories. Ph.D. thesis, Macquarie University, New South Wales (1989)Google Scholar
  7. 7.
    Simpson, C.: A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert–Van Kampen. (Preprint alg-geom/9704006)Google Scholar
  8. 8.
    Street, R.: The algebra of oriented simplexes. J. Pure Appl. Algebra 49(3), 283–335 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tamsamani, Z.: Sur des notions de n-catégorie et n-groupoide non-strictes via des ensembles multi-simpliciaux. K-theory 16(1), 51–99 (1999)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of ChicagoChicagoUSA

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