André Joyal Explained

André Joyal
Birth Date:	25 February 1943
Birth Place:	Drummondville, Quebec, Canada
Fields:	Category theory Homotopy theory
Workplaces:	Université du Québec à Montréal
Known For:	Quasi-categories Combinatorial species

André Joyal (in 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]

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. 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).

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