groupoid | Physics Library

1 Groupoid definitions

Definition 1.1. A groupoid G is a small category in which every morphism is invertible. Its set of objects is denoted by

X = Ob (G).

One often writes G_x^y for the set of morphisms in G from x to y.

Definition 1.2. A topological groupoid consists of a topological space G, a distinguished subspace

G(0) = Ob (G ) ⊆ G,

called the space of objects, together with continuous range and source maps

(0) r,s : G ⇉ G ,

and a continuous law of composition

(2) ∘ : G −→ G,

where

(2) G = G ×G (0) G = {(γ1,γ2) ∈ G × G : s(γ1) = r(γ2 )} .

These data satisfy:

For every (γ₁,γ₂) ∈ G⁽²⁾, $s(γ1 ∘ γ2) = s(γ2), r(γ1 ∘ γ2) = r(γ1).$
For every x ∈ G⁽⁰⁾, $s(x) = r(x) = x.$
For every γ ∈ G, $γ ∘ s(γ) = γ, r(γ) ∘ γ = γ.$
Whenever the products are defined, $(γ1 ∘ γ2) ∘ γ3 = γ1 ∘ (γ2 ∘ γ3).$
Each γ ∈ G has a two-sided inverse γ⁻¹ satisfying $γ ∘ γ− 1 = r (γ ), γ− 1 ∘ γ = s(γ).$
For a topological groupoid, the inverse map is also required to be continuous.

For u ∈ Ob(G), the arrows from u to itself form a group G_u, called the isotropy group of G at u.

Thus, a topological groupoid is a groupoid internal to the category of topological spaces and continuous maps. The notion of an internal groupoid is important in many fields, since groupoids generalize bundles of groups, group actions, and equivalence relations.

Several examples of groupoids are:

locally compact groups, transformation groups, and groups in general;
equivalence relations;
tangent bundles;
the tangent groupoid;
holonomy groupoids for foliations;
Poisson groupoids;
graph groupoids.

As a simple example, consider an equivalence relation R on a set X. Then R becomes a groupoid with

−1 (x, y)(y,z) = (x,z), (x, y) = (y,x).

Its object space is the diagonal copy of X in X × X, with

r(x,y) = x, s(x,y) = y.

The set of composable pairs is

R (2) = {((x,y),(y,z)) : (x,y),(y,z) ∈ R }.

When R = X × X, this is the pair groupoid on X. For

X = {1,2,...,n },

the corresponding pair groupoid is

R = {1,2,...,n } × {1,2,...,n}. n

Identifying (i,j) ∈ R_n with the matrix unit e_ij, the groupoid composition agrees with matrix-unit multiplication whenever the product is defined:

e e = e , (e )−1 = e . ij jk ik ij ji

For a locally compact groupoid G_lc, one usually requires G_lc to be a second-countable locally compact Hausdorff space, with continuous multiplication and inversion.

The analogue of a left Haar measure is a family

{ λu}u∈G(0), lc

where each λ^u is a positive regular Borel measure on the corresponding range fiber. The family is required to vary continuously when integrated against f ∈ C_c(G_lc) and to satisfy left invariance. Such a family is called a left Haar system.

"groupoid" is owned by bci1.
(view preamble)