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André Joyal

André Joyal
Born (1943-02-25) February 25, 1943 (age 81)
Drummondville, Quebec, Canada
Known forQuasi-categories
Combinatorial species
Scientific career
FieldsCategory theory
Homotopy theory
InstitutionsUniversité du Québec à Montréal

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]


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 the category of simplicial sets 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 life

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


  • Joyal, André; Tierney, Myles (1984). "An extension of the Galois theory of Grothendieck". Memoirs of the American Mathematical Society. 51 (309). doi:10.1090/memo/0309. MR 0756176.
  • Joyal, A. (2002). "Quasi-categories and Kan complexes, (in Special volume celebrating the 70th birthday of Prof. Max Kelly)". Journal of Pure and Applied Algebra. 175 (1–3): 207–222. doi:10.1016/S0022-4049(02)00135-4.
  • Joyal, André; Tierney, Myles (2007). "Quasi-categories vs Segal spaces". Categories in Algebra, Geometry and Mathematical Physics. Contemporary Mathematics. Vol. 431. pp. 277–326. arXiv:math/0607820. doi:10.1090/conm/431/08278. ISBN 9780821839706. MR 2342834. S2CID 119749421.
  • Joyal, André; Tierney, Myles (2000). "On the theory of path groupoids". Journal of Pure and Applied Algebra. 149: 69–100. doi:10.1016/S0022-4049(98)00164-9.
  • Joyal, André; Street, Ross (1993). "Pullbacks equivalent to pseudopullbacks" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. 34 (2): 153–156. MR 1223657.
  • Joyal, André; Tierney, Myles (1991). "Strong stacks and classifying spaces". Category Theory. Lecture Notes in Mathematics. Vol. 1488. pp. 213–236. doi:10.1007/BFb0084222. ISBN 978-3-540-54706-8.
  • Joyal, André; Street, Ross (1991). "An introduction to Tannaka duality and quantum groups" (PDF). Category Theory. Lecture Notes in Mathematics. Vol. 1488. pp. 413–492. doi:10.1007/BFb0084235. ISBN 978-3-540-54706-8.
  • Joyal, André; Street, Ross (1991). "The geometry of tensor calculus, I". Advances in Mathematics. 88: 55–112. doi:10.1016/0001-8708(91)90003-P.; Joyal, A.; Street, R. (1993). "Braided Tensor Categories". Advances in Mathematics. 102: 20–78. doi:10.1006/aima.1993.1055.; Joyal, André; Street, Ross (1991). "Tortile Yang-Baxter operators in tensor categories". Journal of Pure and Applied Algebra. 71: 43–51. doi:10.1016/0022-4049(91)90039-5.
  • Joyal, André; Street, Ross; Verity, Dominic (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society. 119 (3): 447–468. doi:10.1017/S0305004100074338. S2CID 50511333.
  • 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. ^ Joyal, André; Tierney, Myles (1984). "An extension of the Galois theory of Grothendieck". Memoirs of the American Mathematical Society. 51 (309). doi:10.1090/MEMO/0309.
  5. ^ A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)
  6. ^ Joyal's CatLab
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André Joyal
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