%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: additive quotient category
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%%% Filename: AdditiveQuotientCategory3.tex
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\subsection{Essential data: Dense subcategory}
\begin{definition}
A full subcategory $\mathcal{A}$ of an \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} $\mathcal{C}$ is called \emph{dense} if for any exact sequence in $\mathcal{C}$:
$$ 0 \to X' \to X \to X'' \to 0,$$
$X$ is in $\mathcal{A}$ if and only if both $X'$ and $X''$ are in $\mathcal{A}$.
\end{definition}
\textbf{Remark 0.1:}
One can readily prove that if $X$ is an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of the \emph{\htmladdnormallink{dense subcategory}{http://planetphysics.us/encyclopedia/DenseSubcategory.html}} $\mathcal{A}$ of
$\mathcal{C}$ as defined above, then any subobject $X_Q$, or \htmladdnormallink{quotient object}{http://planetphysics.us/encyclopedia/DenseSubcategory.html} of $X$, is also in
$\mathcal{A}$. \\
\subsubsection{System of morphisms $\Sigma_A$}
Let $\mathcal{A}$ be a \emph{dense subcategory} (as defined above) of a locally small Abelian category $\mathcal{C}$,
and let us denote by $\Sigma_A$ (or simply only by $\Sigma$ -- when there is no possibility of confusion)
the \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of all \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $s$ of $\mathcal{C}$ such that both $ker s$ and $coker s$ are in $\mathcal{A}$.
One can then prove that the category of additive fractions $\mathcal{C}_{\Sigma}$ of $\mathcal{C}$
relative to $\Sigma$ exists.
\begin{definition}
The \emph{quotient category of $\mathcal{C}$ relative to $\mathcal{A}$}, denoted as $\mathcal{C}/\mathcal{A}$, is defined as the category of additive fractions $\mathcal{C}_{\Sigma}$ relative to a class of morphisms
$\Sigma :=\Sigma_A $ in $\mathcal{C}$.
\end{definition}
\textbf{Remark 0.2}
In view of the restriction to additive fractions in the above definition, it may be more
appropriate to call the above \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{C}/\mathcal{A}$ an \emph{additive quotient category}.
This would be important in order to avoid confusion with the more general notion of
\htmladdnormallink{quotient category}{http://planetphysics.us/encyclopedia/QuotientCategory2.html}--which is defined as a category of fractions. Note however that \textbf{Remark 0.1} is also applicable in the context of the more \htmladdnormallink{general definition}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of a \htmladdnormallink{quotient category}{http://planetphysics.us/encyclopedia/QuotientCategory2.html}.
\end{document}
