nLab André Joyal

André Joyal is a Canadian mathematician, a professor at Université du Québec à Montréal.

He got his PhD in 1971 from Université de 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 once began a project at joyalscatlab.

webpage, wikipedia

Old webpage.

CV from 1995.

Selected writings

On forcing via classifying toposes and the classifying topos of a localic groupoid:

• André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309 (1984) [ISBN:978-1-4704-0722-3]

  (historical note: according to MR756176 (86d:18002) by Peter Johnstone, the main results of this monograph were obtained by the authors around 1978-1979, the typed version circulated from 1982, and the results influenced the field much before the actual publication)

Introducing the notion of braided monoidal categories:

• André Joyal, Ross Street: Braided monoidal categories, Macquarie Math Reports 85-0067 (1985) [pdf, pdf]

• André Joyal, Ross Street: Braided monoidal categories, Macquarie Math Reports 860081 (1986) [pdf, pdf]

• André Joyal, Ross Street, Braided tensor categories, Adv. Math. 102 (1993) 20-78 [doi:10.1006/aima.1993.1055]

On algebraic set theory:

• 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

On transition systems, bisimulations and open morphisms:

• André Joyal, Mogens Nielsen, Glynn Winskel: Bisimulation from open maps, BRICS report series RS-94-7 (1994) [doi:10.7146/brics.v1i7.21663, pdf], Information and Computation 127 2 (1996) 164-185 [doi:10.1006/inco.1996.0057]

Introducing the notion of traced monoidal categories:

• André Joyal, Ross Street, Dominic Verity, Traced monoidal categories, Math. Proc. Camb. Phil. Soc. 119 (1996) 447-468 [pdf, doi:10.1017/S0305004100074338]

On the Dwyer-Kan loop groupoid-construction (path-simplicial groupoids):

• André Joyal, Myles Tierney, On the theory of path groupoids, Journal of Pure and Applied Algebra 149 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

On quasi-categories:

• André Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207-222 (special volume celebrating the 70th birthday of Prof. Max Kelly) [doi:10.1016/S0022-4049(02)00135-4]

On the Quillen equivalence between the model categories for quasi-categories and complete Segal spaces:

• Andre Joyal, Myles Tierney, Quasi-categories vs. Segal spaces, in Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics 431 (2007) [arXiv:math/0607820, doi:10.1090/conm/431]

On quasi-categories:

• André Joyal, Notes on quasi-categories (2008) [pdf, pdf]

• André Joyal, The Theory of Quasi-Categories and its Applications, lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf, pdf)

On simplicial homotopy theory, the classical model structure on simplicial sets and the classical model structure on topological spaces:

• André Joyal, Myles Tierney Notes on simplicial homotopy theory, Lecture at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf)

• André Joyal, Myles Tierney An introduction to simplicial homotopy theory, 2009 (web, pdf)

On homotopy theory via category theory:

• André Joyal, Categorical Homotopy Type Theory, MIT Topology Seminar (March 17, 2014)

Proving the Blakers-Massey theorem in any $(\infty,1)$-topos and with the (n-connected, n-truncated) factorization system allowed to be replaced by more general modalities:

• Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, A Generalized Blakers-Massey Theorem, Journal of Topology 13 4 (2020) 1521-1553 $[$arXiv:1703.09050, doi:10.1112/topo.12163$]$

On $(\infty,1)$-toposes understood as logoi:

• Mathieu Anel, André Joyal, Topo-logie, in New Spaces for Mathematics and Physics, Cambridge University Press (2021) 155-257 [doi:10.1017/9781108854429.007, pdf]

See also:

• 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;

• 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 can be found here

• 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.

category: people