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
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
is an isomorphism.
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.
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.
The general definition is Def. 1.67 in:
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)
Last revised on July 9, 2021 at 11:10:27. See the history of this page for a list of all contributions to it.