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

A note on the Wehrheim-Woodward category

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.
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.

[2]

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

[3]

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

[4]

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

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.

[2]

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

[3]

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

[4]

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

[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Fred Brauer. Age-of-infection and the final size relation. Mathematical Biosciences & Engineering, 2008, 5 (4) : 681-690. doi: 10.3934/mbe.2008.5.681

[14]

Mathieu Molitor. On the relation between geometrical quantum mechanics and information geometry. Journal of Geometric Mechanics, 2015, 7 (2) : 169-202. doi: 10.3934/jgm.2015.7.169

[15]

Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems & Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397

[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]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249

[19]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557

[20]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010

2016 Impact Factor: 0.857

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]