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'''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.
'''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.


== Main research ==
== Main research ==


He invented [[Kripke–Joyal semantics]]<ref>Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209--225, Coll. Publ., London, 2006; [http://www.ams.org/mathscinet-getitem?mr=2008m:03047 MR2008m:03047] </ref>, the theory of [[combinatorial species]] and with M. Tierney a generalization of the Galois theory of [[Alexander Grothendieck|Grothendieck]]<ref>A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.</ref> 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<ref>A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)</ref>. 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 <ref>[http://ncatlab.org/joyalscatlab Joyal's CatLab]</ref> on categorical mathematics.
He invented [[Kripke–Joyal semantics]],<ref>Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209--225, Coll. Publ., London, 2006; [http://www.ams.org/mathscinet-getitem?mr=2008m:03047 MR2008m:03047]</ref> the theory of [[combinatorial species]] and with M. Tierney a generalization of the Galois theory of [[Alexander Grothendieck|Grothendieck]]<ref>A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.</ref> 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.<ref>A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)</ref> 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 <ref>[http://ncatlab.org/joyalscatlab Joyal's CatLab]</ref> on categorical mathematics.


===Works on algebraic equations===
===Works on algebraic equations===
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Joyal proved the following theorem in 1967.
Joyal proved the following theorem in 1967.


If <math> P(z)= \sum_{j=0}^{n} a_jz^j </math> is a polynomial of degree ''n'' such that <math> a_n \geq a_{n-1} \geq \cdots \geq a_1 \geq a_0, a_j \in R </math>, then all the zeros of ''P''(''z'') lie in <math> |z| \leq (a_n - a_0 + |a_0| )/ |a_n| </math>.<ref>A.Joyal, G. Labelle and Q.I.Rehman, On the location of zeros of polynomial, Canad. Math. Bull. 10, (1967), 53&ndash;63, [http://www.ams.org/mathscinet-getitem?mr=0213513 MR0213513] </ref>
If <math> P(z)= \sum_{j=0}^{n} a_jz^j </math> is a polynomial of degree ''n'' such that <math> a_n \geq a_{n-1} \geq \cdots \geq a_1 \geq a_0, a_j \in R </math>, then all the zeros of ''P''(''z'') lie in <math> |z| \leq (a_n - a_0 + |a_0| )/ |a_n| </math>.<ref>A.Joyal, G. Labelle and Q.I.Rehman, On the location of zeros of polynomial, Canad. Math. Bull. 10, (1967), 53&ndash;63, [http://www.ams.org/mathscinet-getitem?mr=0213513 MR0213513]</ref>


==References==
==References==
{{Reflist}}
{{Reflist}}


* 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 [http://dx.doi.org/10.1016/S0022-4049%2802%2900135-4 doi].
* 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 [http://dx.doi.org/10.1016/S0022-4049%2802%2900135-4 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. [http://arxiv.org/abs/math.AT/0607820 math.AT/0607820].
* 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. [http://arxiv.org/abs/math.AT/0607820 math.AT/0607820].


* A. Joyal, M. Tierney, On the theory of path groupoids, J. Pure Appl. Algebra 149 (2000), no. 1, 69--100, [http://dx.doi.org/10.1016/S0022-4049%2898%2900164-9 doi]
* A. Joyal, M. Tierney, On the theory of path groupoids, J. Pure Appl. Algebra 149 (2000), no. 1, 69—100, [http://dx.doi.org/10.1016/S0022-4049%2898%2900164-9 doi]


* A. Joyal, R. Street, Pullbacks equivalent to pseudopullbacks, Cahiers topologie et géométrie différentielle catégoriques 34 (1993) 153-156; [http://www.numdam.org/item?id=CTGDC_1993__34_2_153_0 numdam] MR94a:18004.
* A. Joyal, R. Street, Pullbacks equivalent to pseudopullbacks, Cahiers topologie et géométrie différentielle catégoriques 34 (1993) 153-156; [http://www.numdam.org/item?id=CTGDC_1993__34_2_153_0 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, 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 [http://www.math.mq.edu.au/~street/CT90Como.pdf pdf].
* 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 [http://www.math.mq.edu.au/~street/CT90Como.pdf pdf].


* A. Joyal, R. Street, The geometry of tensor calculus I, Adv. Math. 88(1991), no. 1, 55--112, [http://dx.doi.org/10.1016/0001-8708%2891%2990003-P doi]; Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), no. 1, 43--51, [http://dx.doi.org/10.1016/0022-4049%2891%2990039-5 doi]; Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20--78, [http://dx.doi.org/10.1006/aima.1993.1055 doi].
* A. Joyal, R. Street, The geometry of tensor calculus I, Adv. Math. 88(1991), no. 1, 55—112, [http://dx.doi.org/10.1016/0001-8708%2891%2990003-P doi]; Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), no. 1, 43—51, [http://dx.doi.org/10.1016/0022-4049%2891%2990039-5 doi]; Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20—78, [http://dx.doi.org/10.1006/aima.1993.1055 doi].


* A. Joyal, R. Street, D. Verity, Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447--468.
* A. Joyal, R. Street, D. Verity, Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447—468.


* A. Joyal, [[Ieke Moerdijk|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, [[Ieke Moerdijk|I. Moerdijk]], Algebraic set theory. London Mathematical Society Lecture Note Series 220. Cambridge Univ. Press 1995. viii+123 pp.&nbsp;ISBN 0-521-55830-1


* A. Joyal, The theory of quasi-categories and its applications, lectures at CRM Barcelona February 2008, draft [http://www.crm.cat/HigherCategories/hc2.pdf pdf]
* A. Joyal, The theory of quasi-categories and its applications, lectures at CRM Barcelona February 2008, draft [http://www.crm.cat/HigherCategories/hc2.pdf pdf]
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| ALTERNATIVE NAMES =
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION =
| SHORT DESCRIPTION =
| DATE OF BIRTH =
| DATE OF BIRTH = 1943
| PLACE OF BIRTH =
| PLACE OF BIRTH =
| DATE OF DEATH =
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[[Category:Canadian mathematicians]]
[[Category:Canadian mathematicians]]
[[Category:Category theorists]]
[[Category:Category theorists]]


{{canada-scientist-stub}}
{{mathematician-stub}}


[[fr:André Joyal]]
[[fr:André Joyal]]

Revision as of 05:59, 4 January 2011

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.

Main research

He invented Kripke–Joyal semantics,[1] the theory of combinatorial species and with M. Tierney a generalization of the Galois theory of Grothendieck[2] 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.[3] 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 [4] 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 .[5]

References

  1. ^ Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209--225, Coll. Publ., London, 2006; MR2008m:03047
  2. ^ A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.
  3. ^ A. Joyal, A letter to Grothendieck, April 1983 (contains a Quillen model structure on simplicial presheaves)
  4. ^ Joyal's CatLab
  5. ^ 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).

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