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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

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