%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: groupoid homomorphism
%%% Primary Category Code: 02.
%%% Filename: GroupoidHomomorphism.tex
%%% Version: 1
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
Let ${\mathsf{\G}}_1$ and ${\mathsf{\G}}_2$ be two \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} considered as two distinct \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} with all invertible \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between their \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} (or `elements'), respectively, $ x \in Ob({\mathsf{\G}}_1) = {{{\mathsf{\G}}_0}}^1$ and $ y \in Ob({\mathsf{\G}}_2) = {{{\mathsf{\G}}_0}}^2$. A \emph{groupoid homomorphism} is then defined as a \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $h: {\mathsf{\G}}_1 \longrightarrow {\mathsf{\G}}_2$.
A \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} of groupoid homomorphisms is naturally a \htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, and \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html} of groupoid homomorphisms (as defined above by groupoid functors) preserve groupoid structure(s), i.e., both the \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} and the \htmladdnormallink{topological structure}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of groupoids. Thus, in the case of \htmladdnormallink{topological groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}, $\mathsf{G}$, one also has the associated \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space \emph{\htmladdnormallink{homeomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}} that naturally preserve topological structure.
\end{definition}
\textbf{Remark:}
Note that the morphisms in the \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html}, $Grpd$, are, of course, groupoid homomorphisms, and
that groupoid homomorphisms also form (groupoid) \htmladdnormallink{functor categories}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} defined in the standard manner for
categories.
\end{document}
