Definition 0.1. Let 𝒢1 and 𝒢2 be two groupoids considered as two distinct categories with
all invertible morphisms between their objects (or ‘elements’), respectively, x ∈ Ob(𝒢1) = 𝒢01
and y ∈ Ob(𝒢2) = 𝒢02. A groupoid homomorphism is then defined as a functor h : 𝒢
1→𝒢2.
A composition of groupoid homomorphisms is naturally a homomorphism, and natural
transformations of groupoid homomorphisms (as defined above by groupoid functors)
preserve groupoid structure(s), i.e., both the algebraic and the topological structure of
groupoids. Thus, in the case of topological groupoids, G, one also has the associated
topological space homeomorphisms that naturally preserve topological structure.
Remark: Note that the morphisms in the category of groupoids, Grpd, are, of course, groupoid
homomorphisms, and that groupoid homomorphisms also form (groupoid) functor categories
defined in the standard manner for categories.
