In mathematics, more specifically category theory, a generator (otherwise known as a separator), or generating family (resp. separating family) is a collection of objects that "see enough" of the category that their perspective is enough to determine the morphisms in the category.

Definition

is called a generating family.

is a generator is faithful.

The dual of this structure is then referred to as a cogenerator/cogenerating family.

Characterisations

Some older texts may use a different definition by Grothendieck. When he was developing his theory of Grothendieck categories to determine subobjects. However these definitions coincide for many practical applications in particular when working in a topos.

The usage of the word "generator" evokes the idea that we can generate the category using these objects. This is true in the following sense.

If our category is locally small with all small coproducts is generating if and only if the map

That acts as f on the part of the coproduct with index f, is an epimorphism is generating if and only if every object X

If, in addition, every epimorphism in our category is regular this means that every object of our category is generated by colimits. This is the case in an Abelian category for example.

Projective Generators

Projective generators (and their dual injective cogenerators) are often very powerful tools to have when doing algebra in a category. For example, if an abelian category has small coproducts and a compact projective generator P. This is then used to prove Mitchell's embedding theorem. This fact is generally useful as compactness, projectiveness and being a generating family are stable under reasonable sums and taking summands so many categories that aren't equivalent to R-mod can be approximated by R-mod for some R by limiting the "size" of objects and taking the sum of all the "smaller" projective generators

Using the characterisation in terms of projections from coproducts, having a family of projective generators along with the existence of coproducts guarantees the category has "enough projectives".

we get the idea of a Character module to be the circle group we obtain the theory of Pontryagin duality.

Examples

• In the category of abelian groups is a generator (in fact a projective one): If f and g suffices.

• is a cogenerator (in fact an injective one): Supposing f and g disagree at x is a divisible group if it has order n.

• , the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator.

• In the category of sets, any set with at least two elements is a cogenerator.

• In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator.

• Using the Tietze extension theorem one can show that the unit interval is an injective cogenerator in the category of compact Hausdorff spaces.

External links

References