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

$$Hom_C(X,-) : C \to Set$$

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

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

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.

Examples

• A set $X$ is a finitely generated object in Sets 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 a finitely generated module if it is a quotient of a free module with a finite basis.

• finitely generated group

Related concepts

• finitely presentable object

• generators and relations

• finite object

• object of finite type

References

The general definition is Def. 1.67 in:

• Jiří Adámek, Jiří Rosický, Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, number 189, Cambridge University Press 1994 (doi:10.1017/CBO9780511600579)

Further development and connections to locally generated categories in

• Jiří Adámek, Jiří Rosický, What are locally generated categories,?, Proc. Categ. Conf. Como 1990, Lect. Notes in Math. 1488 (1991), 14-19.

• Ivan Di Liberti, Jiří Rosický, Enriched Locally Generated Categories?, (arxiv:2009.10980)