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.
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 (q.v.).
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:
Equivalently, we have that $S$ is a separator if, for every object $X$ in $\mathcal{C}$, morphisms $f:S\rightarrow X$ are jointly epic. 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:
A family $\mathcal{S} = (S_a)_{(a \in A)}$ of objects of a category $\mathcal{C}$ is a separating family or a generating family if:
Assuming again that $\mathcal{C}$ is locally small, we have equivalently that $\mathcal{S}$ is a separating family if the family of hom functors $Hom(S_a,-) \colon \mathcal{C} \to$ Set (for $a \in A$) is jointly faithful.
Since repetition is irrelevant in a separating family, we may also speak of a separating class instead of a separating family.
A separating set is a small separating class.
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.
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 $f h \neq g h$ .
See Definition B2.4.1 in the Elephant.
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.
In $Set^{op}$, the two-element set $2$ is a separator, thus every continuous functor $F:Set\to C$ into a locally small category $C$ has a left adjoint by the special adjoint functor theorem.
On the other hand, the opposite of the category of groups does not admit a separator, since there exists a continuous functor $ML: Group\to Set$ which does not have a left adjoint (see Example 3.1 on the page adjoint functor theorem).
If $C$ is locally small and has all small coproducts, then a set-indexed family $(S_a)_{(a\colon A)}$ is separating if and only if, for every $X\in C$, the canonical morphism
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:
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 strengthened notion is that of 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:
If $C$ is locally small and has all small coproducts, 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.
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 $\varepsilon_A$.
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.
Assume that $C$ is locally small and has all small coproducts.
If $C$ is balanced, then every separator is extremal.
If $C$ has pullbacks, then every extremal separator is strong.
If $C$ is regular, then every strong separator is regular.
The converse implications do always hold.
This is a direct consequence of the facts that
in a balanced category every epi is extremal,
in a category with pullbacks, every extremal epi is strong, and
in a regular category every strong epi is regular.
Proposition 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.
Most of the literature uses the term “strong separator” (or strong generator) for what we call an extremal separator. 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).
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).
Finally, the strongest kind of separator commonly seen is that of dense separator.
A dense separator in a category $C$ is a family $(S_i)_{(i\colon I)}$ of objects such that the generated full subcategory is dense.
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 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.
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).
Francis Borceux, Handbook of Categorical Algebra I, Cambridge University Press, 1994
Jiří Adámek and Jiří Rosicky, Locally presentable and accessible categories, Cambridge University Press
Last revised on October 18, 2023 at 11:55:07. See the history of this page for a list of all contributions to it.