Categories of (co)isotropic linear relations

TL;DR

This paper explores the structure of categories of linear relations, especially (co)isotropic relations in symplectic vector spaces, and how Wehrheim-Woodward's construction extends these categories while preserving their algebraic properties.

Contribution

It analyzes Wehrheim-Woodward's construction applied to categories of linear and (co)isotropic relations, revealing their structure as central extensions by endomorphisms.

Findings

Wehrheim-Woodward categories are central extensions of original relation categories.

The endomorphisms form a free submonoid with two generators.

The construction preserves categorical structure under transversality and monicity conditions.

Abstract

In categories of linear relations between finite dimensional vector spaces, composition is well-behaved only at pairs of relations satisfying transversality and monicity conditions. A construction of Wehrheim and Woodward makes it possible to impose these conditions while retaining the structure of a category. We analyze the resulting category in the case of all linear relations, as well as for (co)isotropic relations between symplectic vector spaces. In each case, the Wehrheim-Woodward category is a central extension of the original category of relations by the endomorphisms of the unit object, which is a free submonoid with two generators in the additive monoid of pairs of nonnegative integers.