# What is a separator object?

Let $S$ be an object of category $C$. We say $S$ is a separator object of $C$ if whenever $$Y \stackrel{f_1} \longleftarrow X \stackrel{f_2}{\longrightarrow} Y$$ $(\forall x[S\stackrel{x}\longrightarrow X \Rightarrow f_1x=f_2x]) \Rightarrow f_1 =f_2$

This reminds of the definition of an epimorphism. I am wondering if we could define separator object as an object which all of its morphisms are epic.

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It would seem that the outbound morphisms of a separator object are "collectively" right-cancellative, but not necessarily individually right-cancellative. What if $f_1x=f_2x$ but $f_1y\ne f_2y$ and $f_1\ne f_2$ for some morphisms $S\xrightarrow{x}X$ and $S\xrightarrow{y}X$? The definition does not preclude such a situation, but such a situation precludes $x$'s being epic. – anon Jul 15 '12 at 22:58
The point of this definition is that $\text{Hom}(S, -)$ is a faithful functor. – Qiaochu Yuan Jul 16 '12 at 0:29

The definition of a separator object appears to enshrine "collective" right-cancellation, but not the individual right-cancellation that is mandated by epicness. IOW, an object is a separator object when $f_1x=f_2x$ for specific $f_1,f_2$ can be right-cancelled if it holds for all $x$, whereas a specific $x$ is epic if $f_1x=f_2x$ can be right-cancelled for any $f_1,f_2$. These do not mean the same thing (which arrows are being universally quantified vs. individually specified is distinct) and aren't necessarily compatible, so we can't define a separator object as one with all outbound morphisms epic.
Consider the following situation: given $S$, there exist arrows from $S$ to $X$, $S\xrightarrow{x}X$ and $S\xrightarrow{y}X$ , and arrows from $X$ to $Y$, $X\xrightarrow{f}Y$ and $X\xrightarrow{g}Y$, such that $fx=gx$ but $fy\ne gy$ and $f\ne g$. This is not in contradiction with the definition of a separator object (so $S$ could still be a separator), while it does preclude $x$ from being an epimorphism. We can even devise a category with precisely these objects and arrows (plus $1_S,1_X,1_Y$), and $S$ will be a separator because the definition is vacuously fulfilled.
On the other hand, if an object $S$ has the property that all outbound morphisms are epic, then it will also be a separator object, so they are related.
The synonymous term generator (or, more generally, generating families) are about as often encountered. Some easy examples may help to see what is going on: a) A one point set or a one point space. b) $\mathbb{Z}$ as a group or as an abelian group c) the ground ring as a module. d) a group $G$ as a $G$-set. The reason for the "generator" terminology is that, if you have coproducts, every object is an epimorphic image of a coproduct of the generator. – t.b. Jul 15 '12 at 23:35
I probably should have made this explicit... Let $G$ be a generator and let $X$ be an arbitrary object. Assuming that arbitrary coproducts exist, Exercise 0 about generators is that the natural map $\coprod_{f \in \operatorname{Hom}(G,X)} G \to X$ coming from the universal property of the coproduct is an epimorphism. – t.b. Jul 15 '12 at 23:52