groupoid action (original) (raw)

Definition 0.1.

Let 𝒢 be a groupoid and X a topological space. A groupoid action, or 𝒢-action, on X is given by two maps: the anchor map π:X⟶G0 and a map μ:X×G0G1⟶X, with the latter being defined on pairs (x,g) such that π⁢(x)=t⁢(g), written as μ⁢(x,g)=x⁢g. The two maps are subject to the following conditions:

• •
  π⁢(x⁢g)=s⁢(g),
• •
  x⁢u⁢(π⁢(x))=x, and
• •
  (x⁢g)⁢h=x⁢(g⁢h), whenever the operations are defined.

Note:The groupoid action generalizes the concept of group action in a non-trivial way.