nLab
finitely generated object

Definition in arbitrary categories

Let $C$ be a locally small category that admits filtered colimits of monomorphisms. Then an object $X\in C$ is finitely generated if the corepresentable functor

$${\mathrm{Hom}}_C(X,-):C\to \mathrm{Set}$$Hom_C(X,-) : C \to Set

preserves these filtered colimits of monomorphisms. This means that for every filtered category $D$ and every functor $F:D\to C$ such that $F(f)$ is a monomorphism for each morphism $f$ of $D$, the canonical morphism

$$\underset{{\to }_d}{\lim }C(X,F(d))\stackrel{\simeq }{\to }C(X,\underset{{\to }_d}{\lim }F(d))$$\underset{\to_d}{\lim} C(X,F(d)) \stackrel{\simeq}{\to} C(X, \underset{\to_d}{\lim} F(d))

is an isomorphism.

Definition in concrete categories

An object $A$ of a concrete category $C$ is finitely generated if it is a quotient object (in the sense of a regular epimorphism) of some free object $F$ in $C$, where $F$ is free on a finite set.

The object $A$ is finitely presented if it is the coequalizer of a parallel pair $R\rightrightarrows F$ such that $R$ is also free on a finite set.

Related concepts

• finitely presentable object
• generators and relations

Examples

• A set $X$ is a finitely generated object in Set iff it is (Kuratowski-)finite. For this to hold constructively, filtered categories (appearing in the definition of filtered colimit) have to be understood as categories admitting cocones of every Bishop-finite diagram.

• For $R$ a ring, an $R$-module $N$ is finitely generated if it is a quotient of a free module with a finite basis.

References

The general definition is in Locally Presentable and Accessible Categories, definition 1.67.