category theory

# Finitely generated objects

## 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.

## References

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

Revised on November 24, 2013 23:56:43 by Urs Schreiber (89.204.137.196)