%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: fully faithful functor
%%% Primary Category Code: 00.
%%% Filename: FullyFaithfulFunctor2.tex
%%% Version: 4
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
Let $\mathcal{A}$ and $\mathcal{B}$ be two \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} and
let $F: \mathcal{A} \to \mathcal{B}$ be a \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. $F$ is said to be a
{\em fully faithful functor} if it is an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} on
every set $Hom(-,-)$ of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, and that it is {\em essentially \htmladdnormallink{surjective}{http://planetphysics.us/encyclopedia/BCConjecture.html}} if for every \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $X \in \mathcal{B}$, there is some $Y \in \mathcal{A}$ such that $X$ and $F(Y)$ are isomorphic.
\end{definition}
\end{document}
