nLab
generator

Definition

A generator in a category $C$ is an object $S$ such that the functor $h^S=C(S,-):C\to \mathrm{Set}$ is faithful. This means that for any pair $f_1,f_2\in C(X,Y)$, if they are indistinguishable by morphisms from $S$ in the sense that

$$\forall (\theta :S\to X),f_1\circ \theta =f_2\circ \theta ,$$\forall (\theta: S \to X),\; f_1 \circ \theta = f_2 \circ \theta ,

then $f_1=f_2$.

One often extends this notion to a generating family of objects, which is a (usually small) set $𝒮=\{S_a,a\in A\}$ of objects in $C$ such that the family $C(S_a,-)$ is jointly faithful. This means that for any pair $f_1,f_2\in C(X,Y)$, if they are indistinguishable by morphisms from $𝒮$ in the sense that

$$\forall (a:A),\forall (\theta :S_a\to X),f_1\circ \theta =f_2\circ \theta ,$$\forall (a: A),\; \forall (\theta: S_a \to X),\; f_1 \circ \theta = f_2 \circ \theta ,

then $f_1=f_2$.

The dual notion is cogenerator.

Examples and applications

In Set, the point is a generator. More generally, in any well-pointed category, $1$ is a generator. More generally still, in any concrete category, the representing object is a generator.

The standard example of a generator in the category of $R$-modules over a ring $R$ is any free module $R^I$ and $R$ in particular. If a generator is a finitely generated 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 generating family is one of the conditions in Giraud's theorem characterizing Grothendieck toposes.

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

Strengthened generators

If $C$ is locally small and has small coproducts, then a family $\{S_a{\}}_{a\in A}$ is a generating family if and only if for every $X\in C$, the canonical morphism

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

is an epimorphism.

More generally, if $ℰ$ is a subclass of epimorphisms, we say that $\{S_a\}$ is an $ℰ$-generator if each morphism ${\epsilon }_X$ is in $ℰ$.

Of particular importance is the notion of 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 to coproducts, by saying that the family $C(S_a,-)$ is jointly faithful and jointly conservative.

If the inclusion of the full subcategory on the objects $\{S_a\}$ is dense, sometimes $\{S_a\}$ is said to be a dense generator. This is the strongest sort of generator.