category theory

# Separators

## Idea

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

The dual notion is that of a coseparator.

## Caution on terminology

The term ‘generator’ is more ambiguous because of the use of ‘generators’ as in generators and relations.

## Definitions

###### Definition

An object $S\in 𝒞$ of a category $𝒞$ is called a separator or a generator or a separating object or a generating object, or is said to separate morphisms if: * for every pair of parallel morphisms $f,g:X\to Y$ in $𝒞$, if * for every morphism $e:S\to X$, * $u\circ f=u\circ g$, * then $f=g$.

Assuming that $𝒞$ is locally small category, we have equivalently say that $S$ is a separator if the hom functor $\mathrm{Hom}\left(S,-\right):𝒞\to$ Set is faithful.

More generally:

###### Definition

A family $𝒮=\left({S}_{a};\mid ;a:A\right)$ of objects of a category $𝒞$ is a separating family or a generating family if:

• for every pair of parallel morphisms $f,g:X\to Y$ in $𝒞$, if

• for every index $a$ and every morphism $e:{S}_{a}\to X$,

• $e\circ f=e\circ g$, then $f=g$.

Assuming again that $𝒞$ is locally small, we have equivalently say that $𝒮$ is a separating family if the family of hom functors $\mathrm{Hom}\left({U}_{a},-\right):𝒞\to$ Set is jointly faithful?.

Since repetition is irrelevant in a separating family, we may also speak of a separating class instead of a separating family.

###### Definition

A separating set is a small separating class.

## Examples and applications

• In Set, any inhabited set is a separator; in particular, the point is a separator.

• 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.

• 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 separated projective object in the category of $R$-modules, then the traditional terminology is progenerator. Progenerators are important in classical Morita theory, see Morita equivalence.

Mike: The term “progenerator” seems unfortunate to me; it sounds to me like a pro-object that is a generator. Is it well-established? I’ve never heard it, though I have heard “projective generator” in the context of Morita theory.

Zoran Škoda: It is an extremely frequent term in classical algebra and in many of the standard monographs in module theory over classical rings. I personally never use the expression and mentioned it only once in a survey. But as a link to that area of mathematics I tend to behave conservatively. Notice that the terminology subsumes finite generation.

• The existence of a small separating family is one of the conditions in Giraud's theorem characterizing Grothendieck toposes.

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

## Strengthened separators

###### Motivating theorem

If $C$ is locally small and has all small coproducts, then a family $\left({S}_{a}{\right)}_{\left(a:A\right)}$ is a separating family if and only if, for every $X\in C$, the canonical morphism

${\epsilon }_{X}:\coprod _{a:A,f:{S}_{a}\to X}{S}_{a}⟶X$\varepsilon_X\colon \coprod_{a\colon A, f\colon S_a \to X} S_a \longrightarrow X

is an epimorphism.

More generally:

###### Definition

If $ℰ$ is a subclass of epimorphisms, we say that $\left({S}_{a}{\right)}_{\left(a:A\right)}$ is an $ℰ$-separator or $ℰ$-generator if each morphism ${\epsilon }_{X}$ (as above) is in $ℰ$.

Of particular importance is the notion of strong separator or strong generator, which is obtained by taking $ℰ$ to be the class of strong epimorphisms. This can be expressed equivalently, without requiring local smallness or the existence of coproducts, by saying that the family $C\left({S}_{a},-\right)$ is jointly faithful and jointly conservative.

If we take $ℰ$ to be the class of extremal epimorphisms, we might call the resulting notion “extremal separator,” but dense separator or dense generator is more standard. The reason is that the family $\left({S}_{a}{\right)}_{\left(}a:A\right)$ is an extremal separating family if and only if the inclusion of the full subcategory on the objects $\left({S}_{a}{\right)}_{\left(}a:A\right)$ is dense (and this definition makes sense without assuming coproducts or local smallness). This is the strongest sort of separator.

If $C$ has pullbacks, then extremal epis reduce to strong ones, and so extremal separators are necessarily strong. For this reason, some authors simply define “strong generator” to mean dense generator.

Daniel Schaeppi Something seems to be wrong here: strong epimorphisms are extremal, so the notion of extremal generator is weaker than the notion of strong generator. In general, not every strong separator / strong generator is ($\mathrm{Set}$-)dense (take the free abelian group on one separator, for example).

Revised on April 29, 2013 10:18:36 by Tim Porter (95.147.236.157)