A note on the Wehrheim-Woodward category

TL;DR

This paper simplifies the Wehrheim-Woodward category by showing every morphism can be represented by just two relations, a reduction and a coreduction, streamlining the understanding of canonical relations in symplectic geometry.

Contribution

It demonstrates that all morphisms in the Wehrheim-Woodward category can be represented by a sequence of only two relations, simplifying the categorical structure.

Findings

Every morphism is represented by a reduction and a coreduction.

Simplifies the categorical structure of canonical relations.

Provides a minimal representation for morphisms.

Abstract

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.