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 has a separating family if and only if has a small separating family.
A separating family in a fibered category is an object such that for every parallel pair in with and there exist arrows and (constituting a span) such that is -cartesian, and .
See Definition B2.4.1 in the Elephant.
The standard example of a separator in the category of -modules over a ring is any free module (for an inhabited set) and (which is for a point) in particular. If a separator is a finitely generated projective object in the category of -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 (co)separating family is one of the conditions in one version of the adjoint functor theorem.
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 spans or generates if the induced map maps epimorphically onto .
If is a subclass of epimorphisms, we say that is an -separator or -generator if each morphism (as above) is in .
The weakest commonly-seen strengthed generator is an extremal separator.
Slightly stronger is a strong separator or strong generator, which is obtained by taking 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 is jointly faithful and jointly conservative. Since strong epis are extremal, strong generators are extremal.
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 is dense if and only if the functor is full and faithful, where is the inclusion. That is to say, is a dense generator if 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 (-)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.