André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville (formerly Saint-Majorique). He has three children and lives in Montreal.
He discovered Kripke–Joyal semantics, the theory of combinatorial species and with M. Tierney a generalization of the Galois theory of Grothendieck 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 the first real work on quasi-categories, after their invention by Boardman and Vogt, in particular conjecturing 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  on categorical mathematics.
Works on algebraic equations
Joyal proved the following theorem in 1967.
If is a polynomial of degree n such that , then all the zeros of P(z) lie in .
- Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209--225, Coll. Publ., London, 2006; MR2008m:03047
- A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.
- A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)
- Joyal's CatLab
- A.Joyal, G. Labelle and Q.I.Rehman, On the location of zeros of polynomial, Canad. Math. Bull. 10, (1967), 53–63, MR0213513
- 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 space, Category theory (Como, 1990), 213—236, Lecture Notes in Math. 1488, Springer 1991.
- A. 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.
- 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, 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 pdf
- A Joyal, Notes on quasicategories, draft pdf
- A. Joyal, M. Tierney, Notes on simplicial homotopy theory, CRM Barcelona, Jan 2008 pdf
- 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).