category theory

# Separators

## Idea

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

The dual notion is that of a coseparator.

## Caution on 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 1 (q.v.).

## Definitions

###### Definition

An object $S \in \mathcal{C}$ of a category $\mathcal{C}$ 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 $f,g \colon X \to Y$ in $\mathcal{C}$, if $f\circ e = g\circ e$ for every morphism $e\colon S \to X$, then $f = g$.

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

More generally:

###### Definition

A family $\mathcal{S} = (S_a ;|; a\colon A)$ of objects of a category $\mathcal{C}$ is a separating family or a generating family if:

• 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 $f = g$.

Assuming again that $\mathcal{C}$ is 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?.

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 generated projective object in the category of $R$-modules, then one sometimes says (especially in the older literature, e.g. Freyd’s Abelian Categories) that the separator is a progenerator. Progenerators are important in classical Morita theory, see Morita equivalence.

• 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 set $(S_a)_{(a\colon A)}$ is a separating set if and only if, for every $X\in C$, the canonical morphism

$\varepsilon_X\colon \coprod_{a\colon A, f\colon S_a \to X} S_a \longrightarrow X$

is an epimorphism.

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:

###### Definition

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

The weakest commonly-seen strengthed generator is an extremal separator.

Slightly stronger is a strong separator or strong generator, which is obtained by taking $\mathcal{E}$ 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 $C(S_a,-)$ is jointly faithful and jointly conservative. Since strong epis are extremal, strong generators are extremal.

###### Warning

Confusingly, some authors use “strong generator” for what we call an extremal separator. In a category with pullbacks, extremal epis reduce to strong ones, and so extremal separators are necessarily strong, and the clash of terminology is resolved.

Stronger still is a regular generator. Since regular epis are strong, regular generators are strong.

Finally, the strongest sort of generator commonly seen is a dense generator. Dense generators don’t fit into our scheme based on classes of epimorphisms, but they do admit a nice functorial definition: a full subcategory $\mathcal{G} \subset \mathcal{C}$ is dense if and only if the functor $\mathcal{C}(i_{\mathcal{G}}, -): \mathcal{C} \to [\mathcal{G}^{\mathrm{op}}, \mathrm{Set}]$ is full and faithful, where $i_{\mathcal{G}}: \mathcal{G} \to \mathcal{C}$ is the inclusion. That is to say, $\mathcal{G}$ is a dense generator if $i_{\mathcal{G}}$ is a dense functor. In a category with coproducts, every dense generator is regular: this can be seen by reformulating denseness in terms of canonical colimits and expressing the relevant colimit as a coequalizer of two coproducts.

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 ($Set$-)dense (take the free abelian group on one separator, for example).

Tim Campion: I’ve attempted to fix these errors. Hopefully it’s all right now.

Revised on February 28, 2014 01:22:33 by Anonymous Coward (98.223.187.122)