Remarks: The locally compact Hausdorff second countable spaces are analytic. One can therefore say also that G_lc is analytic. When the groupoid G_lc has only one object in its object space, that is, when it becomes a group, the above definition is restricted to that of a locally compact topological group; it is then a special case of a one-object category with all of its morphisms being invertible, that is also endowed with a locally compact, topological structure.

Let us also recall the related concepts of groupoid and topological groupoid, together with the appropriate notations needed to define a locally compact groupoid.

Groupoids and Topological Groupoids

Recall that a groupoid G is a small category with inverses over its set of objects X = Ob(G) . One writes G_x^y for the set of morphisms in G from x to y . A topological groupoid consists of a space G, a distinguished subspace G⁽⁰⁾ = Ob(G) ⊂ G, called the space of objects of G, together with maps

(0.1)

called the range and source maps respectively, together with a law of composition

(0.2)

such that the following hold : 

(1)

s(γ₁ ∘ γ₂) = r(γ₂) , r(γ₁ ∘ γ₂) = r(γ₁) , for all (γ₁,γ₂) ∈ G⁽²⁾ .

(2)

s(x) = r(x) = x , for all x ∈ G⁽⁰⁾ .

(3)

γ ∘ s(γ) = γ , r(γ) ∘ γ = γ , for all γ ∈ G .

(4)

(γ₁ ∘ γ₂) ∘ γ₃ = γ₁ ∘ (γ₂ ∘ γ₃) .

(5)

Each γ has a two–sided inverse γ⁻¹ with γγ⁻¹ = r(γ) , γ⁻¹γ = s(γ) .

Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call G⁽⁰⁾ = Ob(G) the set of objects of G . For u ∈ Ob(G), the set of arrows u→u forms a group G_u, called the isotropy group of G at u.

Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalize bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to ref. [1].

References