André Joyal Explained

André Joyal (pronounced as /fr/; 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]

Research

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, which is now known as Joyal model structure. 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.

Bibliography

• 10.1090/memo/0309. 0756176. An extension of the Galois theory of Grothendieck . 1984 . Joyal . André . Tierney . Myles . Myles Tierney . . 51 . 309 . free .
• 10.1016/S0022-4049(02)00135-4. Quasi-categories and Kan complexes, (in Special volume celebrating the 70th birthday of Prof. Max Kelly). 2002 . Joyal . A. . . 175 . 1–3 . 207–222 .
• Book: 10.1090/conm/431/08278. Quasi-categories vs Segal spaces . Categories in Algebra, Geometry and Mathematical Physics . Contemporary Mathematics . 2007 . Joyal . André . Tierney . Myles . 431 . 277–326 . math/0607820 . 9780821839706 . 119749421. 2342834 .
• 10.1016/S0022-4049(98)00164-9. On the theory of path groupoids . 2000 . Joyal . André . Tierney . Myles . . 149 . 69–100 .
• Pullbacks equivalent to pseudopullbacks . Cahiers de Topologie et Géométrie Différentielle Catégoriques . 1993 . 34 . 2 . 153–156 . Joyal . André . Street . Ross. Ross Street. 1223657.
• Book: 10.1007/BFb0084222. Strong stacks and classifying spaces . Category Theory . Lecture Notes in Mathematics . 1991 . Joyal . André . Tierney . Myles . 1488 . 213–236 . 978-3-540-54706-8 .
• Book: 10.1007/BFb0084235. An introduction to Tannaka duality and quantum groups . http://www.math.mq.edu.au/~street/CT90Como.pdf. Category Theory . Lecture Notes in Mathematics . 1991 . Joyal . André . Street . Ross . 1488 . 413–492 . 978-3-540-54706-8 .
• 10.1016/0001-8708(91)90003-P. The geometry of tensor calculus, I . 1991 . Joyal . André . Street . Ross . . 88 . 55–112 .

10.1006/aima.1993.1055. Braided Tensor Categories . 1993 . Joyal . A. . Street . R. . . 102 . 20–78 . free . ; 10.1016/0022-4049(91)90039-5. Tortile Yang-Baxter operators in tensor categories . 1991 . Joyal . André . Street . Ross . . 71 . 43–51 . free .

• 10.1017/S0305004100074338. Traced monoidal categories . 1996 . Joyal . André . Street . Ross . Verity . Dominic . . 119 . 3 . 447–468 . 50511333 .
• André Joyal, Ieke Moerdijk, Algebraic set theory. London Mathematical Society Lecture Note Series 220. Cambridge Univ. Press 1995. viii+123 pp. 
• 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, 10.2).

See also

• Joyal's theorem
• Joyal's theta category

External links

• Interview with André Joyal (in French)
  • Official Web page at UQAM

Notes and References

1. http://www.ias.edu/people/cos/users/jandre Institute for Advanced Study: A Community of Scholars
2. https://www.math.ias.edu/sp/univalent 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;
4. 10.1090/MEMO/0309. An extension of the Galois theory of Grothendieck . 1984 . Joyal . André . Tierney . Myles . . 51 . 309 . free .
5. A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)
6. https://ncatlab.org/joyalscatlab/published Joyal's CatLab