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André Joyal (French: [ʒwajal]; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013,[1] where he was invited to join the Special Year on Univalent Foundations of Mathematics.[2]

André Joyal
Joyal Andre.jpg
Born (1943-02-25) February 25, 1943 (age 76)
Known forQuasi-categories
Combinatorial species
Scientific career
FieldsCategory theory
Homotopy theory
InstitutionsUniversité du Québec à Montréal



He discovered Kripke–Joyal semantics,[3] the theory of combinatorial species and with Myles Tierney a generalization of the Galois theory of Alexander Grothendieck[4] in the setup of locales. Most of his research is in some way related to category theory, higher category theory and their applications. He did some work on quasi-categories, after their invention by Michael Boardman and Rainer Vogt, in particular conjecturing[5] and proving the existence of a Quillen model structure on sSet whose weak equivalences generalize both equivalence of categories and Kan equivalence of spaces. He co-authored the book "Algebraic Set Theory" with Ieke Moerdijk and recently started a web-based expositional project Joyal's CatLab [6] on categorical mathematics.

Personal lifeEdit

Joyal was born in Drummondville (formerly Saint-Majorique). He has three children and lives in Montreal.


  • André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematical Society 51 (1984), no. 309. doi:10.1090/memo/0309 MR0756176
  • André Joyal, Quasi-categories and Kan complexes, (in Special volume celebrating the 70th birthday of Prof. Max Kelly) J. Pure Appl. Algebra 175 (2002), no. 1-3, 207—222 doi:10.1016/S0022-4049(02)00135-4.
  • André Joyal, Myles Tierney, Quasi-categories vs Segal spaces, Categories in algebra, geometry and mathematical physics, 277—326, Contemp. Math. 431, Amer. Math. Soc., Providence, RI, 2007. arXiv:math.AT/0607820.
  • André Joyal, Myles Tierney, On the theory of path groupoids, J. Pure Appl. Algebra 149 (2000), no. 1, 69—100, doi:10.1016/S0022-4049(98)00164-9.
  • André Joyal, Ross Street, Pullbacks equivalent to pseudopullbacks, Cahiers topologie et géométrie différentielle catégoriques 34 (1993) 153-156; numdam MR1223657.
  • André Joyal, Myles Tierney, Strong stacks and classifying space, Category theory (Como, 1990), 213—236, Lecture Notes in Math. 1488, Springer 1991.
  • André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, Category theory (Como, 1990), 413—492, Lecture Notes in Math. 1488, Springer 1991 pdf.
  • André Joyal, Ross Street, The geometry of tensor calculus I, Adv. Math. 88(1991), no. 1, 55—112, doi:10.1016/0001-8708(91)90003-P; Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), no. 1, 43—51, doi:10.1016/0022-4049(91)90039-5; Braided tensor categories, Advances in Mathematics 102 (1993), no. 1, 20—78, doi:10.1006/aima.1993.1055.
  • André Joyal, Ross Street, Dominic Verity, Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447—468.
  • André Joyal, Ieke Moerdijk, Algebraic set theory. London Mathematical Society Lecture Note Series 220. Cambridge Univ. Press 1995. viii+123 pp. ISBN 0-521-55830-1
  • André Joyal, Myles Tierney, Notes on simplicial homotopy theory, CRM Barcelona, Jan 2008 pdf
  • André Joyal, Disks, duality and theta-categories, preprint (1997) (contains an original definition of a weak n-category: for a short account see Leinster's arXiv:math.CT/0305049, 10.2).


  1. ^ Institute for Advanced Study: A Community of Scholars
  2. ^ IAS school of mathematics: Univalent Foundations of Mathematics
  3. ^ Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209--225, Coll. Publ., London, 2006; MR2396933
  4. ^ A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematical Society 51 (1984), no. 309, vii+71 pp.
  5. ^ A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)
  6. ^ Joyal's CatLab

External linksEdit