June  2018, 10(2): 217-250. doi: 10.3934/jgm.2018009

Double groupoids and the symplectic category

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

Received  October 2017 Revised  December 2017 Published  May 2018

We introduce the notion of a symplectic hopfoid, a "groupoid-like" object in the category of symplectic manifolds whose morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on a new realization of the core of a symplectic double groupoid as a symplectic quotient of the total space. The resulting constructions apply more generally to give a correspondence between double Lie groupoids and groupoid-like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions.

Citation: Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
References:
[1]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.  Google Scholar

[2]

A. Cattaneo and I. Contreras, Relational symplectic groupoids, Letters in Mathematical Physics, 105 (2015), 723-767.  doi: 10.1007/s11005-015-0760-3.  Google Scholar

[3]

A. CattaneoB. Dherin and A. Weinstein, Symplectic microgeometry I: Micromorphisms, Journal of Symplectic Geometry, 8 (2010), 205-223.   Google Scholar

[4]

A. CosteP. Dazord and A. Weinstein, Groupoïdes symplectiques, Publications du Départment de mathématiques, Nouvelle Série. A, 2 (1987), 1-62.   Google Scholar

[5]

R. Hepworth, Vector fields and flows on differential stacks, Theory and Applications of Categories, 22 (2009), 542-587.   Google Scholar

[6]

D. Li-Bland and A. Weinstein, Selective categories and linear canonical relations, SIGMA, 10 (2014), Paper 100, 31 pp. doi: 10.3842/SIGMA.2014.100.  Google Scholar

[7]

K. C. H. Mackenzie, On symplectic double groupoids and the duality of Poisson groupoids, International Journal of Mathematics, 10 (1999), 435-456.  doi: 10.1142/S0129167X99000185.  Google Scholar

[8]

R. A. Mehta and X. Tang, From double Lie groupoids to local Lie 2-groupoids, Bulletin of the Brazilian Mathematical Society, 42 (2011), 651-681.  doi: 10.1007/s00574-011-0033-4.  Google Scholar

[9]

I. Szymczak and S. Zakrzewski, Quantum deformations of the Heisenberg group obtained by geometric quantization, Journal of Geometry and Physics, 7 (1990), 553-569.  doi: 10.1016/0393-0440(90)90006-O.  Google Scholar

[10]

A. Weinstein, The volume of a differentiable stack, Lett Math Phys, 90 (2009), 353-371.  doi: 10.1007/s11005-009-0343-2.  Google Scholar

[11]

A. Weinstein, Coisotropic calculus and Poisson groupoids, Journal of the Mathematical Society of Japan, 40 (1988), 705-727.  doi: 10.2969/jmsj/04040705.  Google Scholar

[12]

A. Weinstein, A note on the Wehrheim-Woodward category, Journal of Geometric Mechanics, 3 (2011), 507-515.  doi: 10.3934/jgm.2011.3.507.  Google Scholar

[13]

A. Weinstein, The symplectic category, Differential Geometric Methods in Mathematical Physics, Lecture Notes in Mathematics, 905 (1982), Springer, Berlin-New York, 45–51.  Google Scholar

[14]

S. Zakrzewski, Quantum and classical pseudogroups. Ⅰ. Union pseudogroups and their quantization, Comm. Math. Phys., 134 (1990), 347-370.   Google Scholar

[15]

S. Zakrzewski, Quantum and Classical Pseudogroups. Ⅱ. Differential and symplectic pseudogroups, Comm. Math. Phys., 134 (1990), 371-395.   Google Scholar

show all references

References:
[1]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.  Google Scholar

[2]

A. Cattaneo and I. Contreras, Relational symplectic groupoids, Letters in Mathematical Physics, 105 (2015), 723-767.  doi: 10.1007/s11005-015-0760-3.  Google Scholar

[3]

A. CattaneoB. Dherin and A. Weinstein, Symplectic microgeometry I: Micromorphisms, Journal of Symplectic Geometry, 8 (2010), 205-223.   Google Scholar

[4]

A. CosteP. Dazord and A. Weinstein, Groupoïdes symplectiques, Publications du Départment de mathématiques, Nouvelle Série. A, 2 (1987), 1-62.   Google Scholar

[5]

R. Hepworth, Vector fields and flows on differential stacks, Theory and Applications of Categories, 22 (2009), 542-587.   Google Scholar

[6]

D. Li-Bland and A. Weinstein, Selective categories and linear canonical relations, SIGMA, 10 (2014), Paper 100, 31 pp. doi: 10.3842/SIGMA.2014.100.  Google Scholar

[7]

K. C. H. Mackenzie, On symplectic double groupoids and the duality of Poisson groupoids, International Journal of Mathematics, 10 (1999), 435-456.  doi: 10.1142/S0129167X99000185.  Google Scholar

[8]

R. A. Mehta and X. Tang, From double Lie groupoids to local Lie 2-groupoids, Bulletin of the Brazilian Mathematical Society, 42 (2011), 651-681.  doi: 10.1007/s00574-011-0033-4.  Google Scholar

[9]

I. Szymczak and S. Zakrzewski, Quantum deformations of the Heisenberg group obtained by geometric quantization, Journal of Geometry and Physics, 7 (1990), 553-569.  doi: 10.1016/0393-0440(90)90006-O.  Google Scholar

[10]

A. Weinstein, The volume of a differentiable stack, Lett Math Phys, 90 (2009), 353-371.  doi: 10.1007/s11005-009-0343-2.  Google Scholar

[11]

A. Weinstein, Coisotropic calculus and Poisson groupoids, Journal of the Mathematical Society of Japan, 40 (1988), 705-727.  doi: 10.2969/jmsj/04040705.  Google Scholar

[12]

A. Weinstein, A note on the Wehrheim-Woodward category, Journal of Geometric Mechanics, 3 (2011), 507-515.  doi: 10.3934/jgm.2011.3.507.  Google Scholar

[13]

A. Weinstein, The symplectic category, Differential Geometric Methods in Mathematical Physics, Lecture Notes in Mathematics, 905 (1982), Springer, Berlin-New York, 45–51.  Google Scholar

[14]

S. Zakrzewski, Quantum and classical pseudogroups. Ⅰ. Union pseudogroups and their quantization, Comm. Math. Phys., 134 (1990), 347-370.   Google Scholar

[15]

S. Zakrzewski, Quantum and Classical Pseudogroups. Ⅱ. Differential and symplectic pseudogroups, Comm. Math. Phys., 134 (1990), 371-395.   Google Scholar

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