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\begin{document}

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\section*{separator}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{separators}{}\section*{{Separators}}\label{separators}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Terminology}{Caution on terminology}\dotfill \pageref*{Terminology} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{fibered}{In fibered categories}\dotfill \pageref*{fibered} \linebreak
\noindent\hyperlink{examples_and_applications}{Examples and applications}\dotfill \pageref*{examples_and_applications} \linebreak
\noindent\hyperlink{strengthened_separators}{Strengthened separators}\dotfill \pageref*{strengthened_separators} \linebreak
\noindent\hyperlink{dense_separators}{Dense separators}\dotfill \pageref*{dense_separators} \linebreak
\noindent\hyperlink{girauds_axioms}{Giraud's axioms}\dotfill \pageref*{girauds_axioms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An [[object]] $S$ (or family $\mathcal{S}$ of objects) in a [[category]] $\mathcal{C}$ is called a \emph{separator} or \emph{generator} if [[generalized elements]] with [[domain]] $S$ (or domain from $\mathcal{S}$) are sufficient to distinguish [[morphisms]] in $\mathcal{C}$.

The [[duality|dual]] notion is that of a \emph{[[coseparator]]}.

\hypertarget{Terminology}{}\subsection*{{Caution on terminology}}\label{Terminology}

The term `generator' is slightly more ambiguous because of the use of `generators' in [[generators and relations]]. That said, there is a link between these two senses provided by theorem \ref{motive} (q.v.).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $S \in \mathcal{C}$ of a [[category]] $\mathcal{C}$ is called a \textbf{separator} or a \textbf{generator} or a \textbf{separating object} or a \textbf{generating object}, or is said to \textbf{separate morphisms} if:

\begin{itemize}%
\item for every pair of [[parallel morphisms]] $f,g \colon X \to Y$ in $\mathcal{C}$, if $f\circ e = g\circ e$ for every morphism $e\colon S \to X$, then $f = g$.

\end{itemize}
\end{defn}
Assuming that $\mathcal{C}$ is [[locally small|locally small category]], we have equivalently that $S$ is a separator if the [[hom functor]] $Hom(S,-) \colon \mathcal{C} \to$ [[Set]] is [[faithful functor|faithful]].

More generally:

\begin{defn}
\label{}\hypertarget{}{}
A [[family]] $\mathcal{S} = (S_a)_{(a\colon A)}$ of [[objects]] of a [[category]] $\mathcal{C}$ is a \textbf{separating family} or a \textbf{generating family} if:

\begin{itemize}%
\item for every pair of [[parallel morphisms]] $f,g \colon X \to Y$ in $\mathcal{C}$, if $f \circ e = g \circ e$ for every $e \colon S_a \to X$ sourced in the family, then $f = g$.

\end{itemize}
\end{defn}
Assuming again that $\mathcal{C}$ is [[locally small category|locally small]], we have equivalently that $\mathcal{S}$ is a separating family if the family of [[hom functors]] $Hom(U_a,-) \colon \mathcal{C} \to$ [[Set]] is [[jointly faithful family of functors|jointly faithful]].

Since repetition is irrelevant in a separating family, we may also speak of a \emph{separating [[class]]} instead of a separating family.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{separating set} is a [[size issues|small]] separating class.

\end{defn}
\hypertarget{fibered}{}\subsubsection*{{In fibered categories}}\label{fibered}

The notion of separating family can be generalized from categories to [[fibered categories]] in such a way that the [[family fibration]] of a category $\mathbf{C}$ has a separating family if and only if $\mathbf{C}$ has a small separating family.

\begin{defn}
\label{}\hypertarget{}{}
A separating family in a [[fibered category]] $P:\mathbf{E}\to \mathbf{B}$ is an object $S\in \mathbf{E}$ such that for every parallel pair $f,g:A\to B$ in $E$ with $f\neq g$ and $P(f) = P(g)$ there exist arrows $c: X\to S$ and $h:X\to A$ (constituting a span) such that $c$ is $P$-cartesian, and $f h \neq g h$ .

\end{defn}
See Definition B2.4.1 in the [[Elephant]].

\hypertarget{examples_and_applications}{}\subsection*{{Examples and applications}}\label{examples_and_applications}

\begin{itemize}%
\item In [[Set]], any [[inhabited set]] is a separator; in particular, the [[point]] is a separator.


\item More generally, in any [[well-pointed category]], any [[terminal object]] is a separator. More generally still, in any represented [[concrete category]], the representing object is a separator.


\item The standard example of a separator in the category of $R$-[[modules]] over a [[ring]] $R$ is any [[free module]] $R^I$ (for $I$ an [[inhabited set]]) and $R$ (which is $R^I$ for $I$ a [[point]]) in particular. If a separator is a finitely generated [[projective object]] in the category of $R$-modules, then one sometimes says (especially in the older literature, e.g. Freyd's \emph{Abelian Categories}) that the separator is a \emph{progenerator}. Progenerators are important in classical Morita theory, see [[Morita equivalence]].


\item The existence of a small separating family is one of the conditions in [[Giraud's theorem]] characterizing [[Grothendieck topos]]es.


\item The existence of a small (co)separating family is one of the conditions in one version of the [[adjoint functor theorem]].



\end{itemize}
\hypertarget{strengthened_separators}{}\subsection*{{Strengthened separators}}\label{strengthened_separators}

\begin{theorem}
\label{motive}\hypertarget{motive}{}
If $C$ is [[locally small category|locally small]] and has all small [[coproduct]]s, then a set-indexed family $(S_a)_{(a\colon A)}$ is separating if and only if, for every $X\in C$, the canonical morphism

\begin{displaymath}
\varepsilon_X\colon \coprod_{a\colon A, f\colon S_a \to X} S_a \longrightarrow X
\end{displaymath}
is an [[epimorphism]].

\end{theorem}
This theorem explains a likely origin of the term ``generator'' or ``generating family''. For example, in linear algebra, one says that a set of morphisms $f_a: S_a \to X$ spans or generates $X$ if the induced map $\oplus S_a \to X$ maps epimorphically onto $X$.

More generally:

\begin{defn}
\label{}\hypertarget{}{}
If $\mathcal{E}$ is a subclass of epimorphisms, we say that $(S_a)_{(a\colon A)}$ is an \emph{$\mathcal{E}$-separator} or \textbf{$\mathcal{E}$-generator} if each morphism $\varepsilon_X$ (as above) is in $\mathcal{E}$.

\end{defn}
The weakest commonly-seen strengthened notion is that of \textbf{extremal separator}, i.e. separator where all maps $\varepsilon_X$ are [[extremal epimorphisms]]. The notion of extremal separator admits an equivalent reformulation not referencing coproducts:

\begin{prop}
\label{Extremal}\hypertarget{Extremal}{}
If $C$ is [[locally small category|locally small]] and has all small [[coproduct]]s, then a set-indexed family $(S_i)_{(i\colon I)}$ is an extremal separator if and only if the functors $C(S_i,-):C\to\mathrm{Set}$ are jointly faithful and jointly [[conservative functor|conservative]].

\end{prop}
\begin{proof}
Assume first that the family $(S_i)_{(i\colon I)}$ is an extremal separator. The functors $C(S_i,-):C\to\mathrm{Set}$ are jointly faithful for every separator. To see that they are also jointly conservative, let $f:A\to B$ such that all $C(S_i,f)$ are bijective. Then $\varepsilon_B$ factors through $f$ since all its components do, which implies that $f$ is an extremal epi since $\varepsilon_B$ is one by assumption. It remains to show that $f$ is a monomorphism. For this, let $u,v:X\to A$ such that $f u = f v$. Then we have $f u h = f v h$ for all $i\in I$ and $h:S_i\to X$, which implies $u h = v h$ since the $C(S_i,f)$ are bijective, and we conclude that $u=v$ since $(S_i)_{(i\colon I)}$ is separating.

Conversely, assume that the functors $C(S_i,-)$ are jointly faithful and jointly conservative. Given $A\in C$, joint faithfulness implies that $\varepsilon_A$ is epic. To see that it is extremally so, assume a factorization $\varepsilon_A = m g$ with $m$ monic. We have to show that $m$ is an isomorphism, and for this it is sufficient to show that all $C(S_i,m)$ are bijections. Injectivity is clear since $m$ is monic, and surjectivity follows since every $h:S_i\to A$ factors through $\varepsilon_A$.

\end{proof}
The concepts ``strong separator'' and ``regular separator'' corresponding to the notions of [[strong epimorphism]] and [[regular epimorphism]] do not admit such a reformulation, but the following result shows that they are equivalent to extremal separators in reasonable categories.

\begin{prop}
\label{}\hypertarget{}{}
Assume that $C$ is [[locally small category|locally small]] and has all small [[coproduct]]s.

\begin{enumerate}%
\item If $C$ is balanced, then every separator is extremal.


\item If $C$ has pullbacks, then every extremal separator is strong.


\item If $C$ is [[regular category|regular]], then every strong separator is regular.



\end{enumerate}
The converse implications do always hold.

\end{prop}
\begin{proof}
This is a direct consequence of the facts that

\begin{enumerate}%
\item in a balanced category every epi is extremal,


\item in a category with pullbacks, every extremal epi is strong, and


\item in a regular category every strong epi is regular.



\end{enumerate}
\end{proof}
\begin{uremark}
\begin{enumerate}%
\item Proposition \ref{Extremal} gives rise to a notion of extremal separator that makes sense independently of the existence of coproducts. In fact claim 1 of the preceding result holds in this more general setting, since every faithful functor out of a balanced category is conservative.


\item Most of the literature uses the term ``strong separator'' (or strong generator) for what we call an extremal separator. \hyperlink{AdamekRosicky}{Adamek and Rosicky (Section 0.6)} also comment on this mismatch, writing ``It would be more reasonable, but unfortunately less standard, to call a strong generator an extremal generator''. However, item 2 of the preceding result shows that this discrepancy disappears and the terms coincide in presence of pullbacks (and coproducts).


\item In the Elephant, Johnstone uses ``separator'' in the same sense as we do, and writes ``generator'' for extremal separators, in the more general sense not assuming coproducts. Since he always assumes finite limits, he can use a simplified criterion only requiring joint conservativity of the hom-functors (since a conservative functor $F:C\to D$ is automatically faithful whenever $C$ has equalizers and $F$ preserves them).



\end{enumerate}
\end{uremark}
\hypertarget{dense_separators}{}\subsubsection*{{Dense separators}}\label{dense_separators}

Finally, the strongest kind of separator commonly seen is that of \textbf{dense separator}.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{dense separator} in a category $C$ is a family $(S_i)_{(i\colon I)}$ of objects such that the generated full subcategory is [[dense subcategory|dense]].

\end{defn}
Every dense separator is an extremal separator, and it is also strong and regular whenever those words make sense, i.e. $C$ is locally small and has small coproducts. If $C$ furthermore has pullbacks and the coproducts are pullback-stable, then every regular separator is dense (see \hyperlink{Borceux}{Borceux I, Proposition 4.5.6}). To see that the pullback-stability condition is necessary, consider the category of abelian groups. Here, the free group on one generator is a regular, but not a dense separator.

\hypertarget{girauds_axioms}{}\subsubsection*{{Giraud's axioms}}\label{girauds_axioms}

[[Grothendieck topos\#Giraud|Giraud's axioms]] characterize Grothendieck toposes as locally small regular categories with effective equivalence relations and disjoint and pullback-stable coproducts admitting a small separator. The previously stated and cited results show that in fact every such separator is dense (the effectivity and disjointness assumptions don't play a role for this conclusion).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[coseparator]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Francis Borceux]], \emph{Handbook of Categorical Algebra I}, Cambridge University Press, 1994


\item [[Jiří Adámek]] and [[Jiří Rosicky]], Locally presentable and accessible categories, Cambridge University Press



\end{itemize}
[[!redirects separator]] [[!redirects separators]]

[[!redirects separating family]] [[!redirects separating families]] [[!redirects generating family]] [[!redirects generating families]]

[[!redirects separating class]] [[!redirects separating classes]] [[!redirects generating class]] [[!redirects generating classes]]

[[!redirects separating set]] [[!redirects separating sets]] [[!redirects generating set]] [[!redirects generating sets]]

[[!redirects strong separator]] [[!redirects strong separators]] [[!redirects strong generator]] [[!redirects strong generators]] [[!redirects strong separating set]] [[!redirects strong separating sets]] [[!redirects strong generating set]] [[!redirects strong generating sets]]

[[!redirects dense separator]] [[!redirects dense separators]] [[!redirects dense generator]] [[!redirects dense generators]]



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