PlanetPhysics/Groupoid Homomorphism

Let ${𝖦}_1$ and ${𝖦}_2$ be two [[../GroupoidHomomorphism2|groupoids]] considered as two distinct [[../Cod|categories]] with all invertible [[../TrivialGroupoid|morphisms]] between their [[../TrivialGroupoid|objects]] (or 'elements'), respectively, $x\in Ob({𝖦}_1)={{𝖦}_0}^1$ and $y\in Ob({𝖦}_2)={{𝖦}_0}^2$. A groupoid homomorphism is then defined as a [[../TrivialGroupoid|functor]] $h:{𝖦}_1⟶{𝖦}_2$.

A [[../Cod|composition]] of groupoid homomorphisms is naturally a [[../TrivialGroupoid|homomorphism]], and [[../VariableCategory2|natural transformations]] of groupoid homomorphisms (as defined above by groupoid functors) preserve groupoid structure(s), i.e., both the [[../CoIntersections|algebraic]] and the [[../TrivialGroupoid|topological structure]] of groupoids. Thus, in the case of [[../GroupoidHomomorphism2|topological groupoids]], $𝖦$, one also has the associated [[../CoIntersections|topological]] space [[../Homeomorphisms|homeomorphisms]] [[../TrivialGroupoid|trivial groupoid]] that naturally preserve topological structure.

Remark: Note that the morphisms in the [[../GroupoidCategory|category of groupoids]], $Grpd$, are, of course, groupoid homomorphisms, and that groupoid homomorphisms also form (groupoid) [[../TrivialGroupoid|functor categories]] defined in the standard manner for categories.