Editing Groupoid (section)

== Groupoids with geometric structures ==

When studying geometrical objects, the arising groupoids often carry a [[Topological space|topology]], turning them into [[topological groupoid]]s, or even some [[differentiable structure]], turning them into [[Lie groupoid]]s. These last objects can be also studied in terms of their associated [[Lie algebroid]]s, in analogy to the relation between [[Lie group]]s and [[Lie algebra]]s.

Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in [[Poisson geometry]] one has the notion of a [[symplectic groupoid]], which is a Lie groupoid endowed with a compatible [[Symplectic manifold|symplectic form]]. Similarly, one can have groupoids with a compatible [[Riemannian metric]], or [[Complex manifold|complex structure]], etc.