American Institute of Mathematical Sciences

Advanced
December  2011, 3(4): 507-515. doi: 10.3934/jgm.2011.3.507

A note on the Wehrheim-Woodward category

Alan Weinstein 1,

1. 

Department of Mathematics, University of California, Berkeley, CA 94720, United States

Received  December 2010 Revised  March 2011 Published  February 2012

Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.
Keywords: canonical relation, categories of relations., Symplectic manifold, lagrangian submanifold.
Mathematics Subject Classification: Primary: 53D12; Secondary: 18B1.
Citation: Alan Weinstein. A note on the Wehrheim-Woodward category. Journal of Geometric Mechanics, 2011, 3 (4) : 507-515. doi: 10.3934/jgm.2011.3.507
References:
[1]

S. Benenti and V. M. Tulczyjew, Relazioni lineari binarie tra spazi vettoriali di dimensione finita,, Memorie Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (5), 3 (1979), 67.   Google Scholar

[2]

M. M. Cohen, "A Course in Simple-Homotopy Theory,'', Graduate Texts in Mathematics, 10 (1973).   Google Scholar

[3]

J. Rognes, Lecture notes on algebraic k-theory,, April 29, (2010).   Google Scholar

[4]

K. Wehrheim and C. T. Woodward, Functoriality for Lagrangian correspondences in Floer theory,, Quantum Topology, 1 (2010), 129.  doi: 10.4171/QT/4.  Google Scholar

show all references

References:
[1]

S. Benenti and V. M. Tulczyjew, Relazioni lineari binarie tra spazi vettoriali di dimensione finita,, Memorie Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (5), 3 (1979), 67.   Google Scholar

[2]

M. M. Cohen, "A Course in Simple-Homotopy Theory,'', Graduate Texts in Mathematics, 10 (1973).   Google Scholar

[3]

J. Rognes, Lecture notes on algebraic k-theory,, April 29, (2010).   Google Scholar

[4]

K. Wehrheim and C. T. Woodward, Functoriality for Lagrangian correspondences in Floer theory,, Quantum Topology, 1 (2010), 129.  doi: 10.4171/QT/4.  Google Scholar

[1]

Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243
[2]

Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811
[3]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
[4]

Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
[5]

Umesh V. Dubey, Vivek M. Mallick. Spectrum of some triangulated categories. Electronic Research Announcements, 2011, 18: 50-53. doi: 10.3934/era.2011.18.50
[6]

G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10.

[7]

Ali Aytekin, Kadir Emir. Colimits of crossed modules in modified categories of interest. Electronic Research Archive, 2020, 28 (3) : 1227-1238. doi: 10.3934/era.2020067
[8]

Uta Renata Freiberg. Einstein relation on fractal objects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509
[9]

Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069
[10]

Nikita Selinger. Topological characterization of canonical Thurston obstructions. Journal of Modern Dynamics, 2013, 7 (1) : 99-117. doi: 10.3934/jmd.2013.7.99
[11]

Peter Scott and Gadde A. Swarup. Regular neighbourhoods and canonical decompositions for groups. Electronic Research Announcements, 2002, 8: 20-28.

[12]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[13]

Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu. A final size relation for epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 159-175. doi: 10.3934/mbe.2007.4.159
[14]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
[15]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[16]

Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253
[17]

Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial & Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611
[18]

Christophe Cheverry, Adrien Fontaine. Dispersion relations in cold magnetized plasmas. Kinetic & Related Models, 2017, 10 (2) : 373-421. doi: 10.3934/krm.2017015
[19]

Artur Babiarz, Adam Czornik, Michał Niezabitowski, Evgenij Barabanov, Aliaksei Vaidzelevich, Alexander Konyukh. Relations between Bohl and general exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5319-5335. doi: 10.3934/dcds.2017231
[20]

Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences