André Joyal is a Canadian mathematician, a professor at Université du Québec à Montréal.
His wide mathematical work, is mainly in category theory, topos theory and abstract homotopy theory. His works include a wide generalization of Galois theory with Myles Tierney, the combinatorial ideas of “Joyal’s species”, discovery of the category structure on the collection of Conway combinatorial games, the discovery of Kripke-Joyal semantics, a series of works (mainly with Ross Street) about (braided, tortile etc.) monoidal categories prompted partly by methods and motivation in theoretical physics, much of his work for about last 30 years centered on developing the theory of quasicategories, after the first ideas of Boardman and Vogt. In the 1980s Joyal invented a Quillen model category structure on the category of simplicial sets (and categories of simplicial presheaves). Joyal and J. Kock more recently proved Simpson's conjecture (on higher categories via weak units) in categorical dimension 3.
Joyal promoted quasi-categories, greatly extending their theory, as a basis for (∞,1)-category theory.
Joyal has contributed to the $n$Lab as ‘joyal’; he is also beginning a new project at joyalscatlab.
A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp. (historical $n$lab note: according to MR756176 (86d:18002) featured review by Peter Johnstone, the main results of this monograph were obtained by the authors around 1978-1979, typed version circulated from 1982, and the results influenced the field much before the actual publication)
A. 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).
A. Joyal, M. Tierney, Quasi-categories vs Segal spaces, Categories in algebra, geometry and mathematical physics, 277–326, Contemp. Math. 431, Amer. Math. Soc., Providence, RI, 2007. math.AT/0607820.
A. Joyal, M. Tierney, On the theory of path groupoids, J. Pure Appl. Algebra 149 (2000), no. 1, 69–100, doi
A. Joyal, R. Street, Pullbacks equivalent to pseudopullbacks, Cahiers topologie et géométrie différentielle catégoriques 34 (1993) 153-156; numdam MR94a:18004.
A. Joyal, M. Tierney, Strong stacks and classifying spaces, Category theory (Como, 1990), 213–236, Lecture Notes in Math. 1488, Springer 1991.
A. Joyal, R. Street, An introduction to Tannaka duality and quantum groups, Category theory (Como, 1990), 413–492, Lecture Notes in Math. 1488, Springer 1991 (pdf).
A. Joyal, R. Street, The geometry of tensor calculus I, Adv. Math. 88(1991), no. 1, 55–112, doi; Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), no. 1, 43–51, doi; Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78, doi.
A. Joyal, R. Street, Braided monoidal categories, Macquarie Math Reports 860081 (1986) pdf; Macquarie Math Reports 850067 (1985) pdf.
A. Joyal, R. Street, D. Verity, Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447–468.
A. Joyal, I. Moerdijk, Algebraic set theory. London Mathematical Society Lecture Note Series 220. Cambridge Univ. Press 1995. viii+123 pp. ISBN: 0-521-55830-1
A. Joyal, The theory of quasi-categories and its applications, lectures at CRM Barcelona February 2008, draft hc2.pdf and pdf at backup location
A Joyal, Notes on quasicategories, (draft)
A. Joyal, M. Tierney, Notes on simplicial homotopy theory, CRM Barcelona, Jan 2008 pdf, backup pdf
A. Joyal, Letter to Alexander Grothendieck, 11.4.1984, Lettre d'Andre Joyal a Alexandre Grothendieck.
A. Joyal, Disks, duality and $\Theta$-categories, preprint (1997) (contains an original definition of a weak $n$-category: for a short account see Leinster’s book, 10.2).
A. Joyal, Remarques sur la théorie des jeux à deux personnes, Gazette des Sciences Mathematiques du Québec 1(4):46–52, 1977; Robin Houston’s rough translation ps
André Joyal, Free lattices, communication and money games, in: Logic and scientific methods. Volume one of the proceedings of the tenth international congress of logic, methodology and philosophy of science, Florence, Italy, Synth. Libr. 259, pages 29–68. Dordrecht: Kluwer Academic Publishers, 1997.