finitely generated object

Finitely generated objects

Definition in arbitrary categories

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

Hom C(X,):CSet Hom_C(X,-) : C \to Set

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

lim dC(X,F(d))C(X,lim dF(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 AA of a concrete category CC is finitely generated if it is a quotient object (in the sense of a regular epimorphism) of some free object FF in CC, where FF is free on a finite set.

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



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

Last revised on November 24, 2013 at 23:56:43. See the history of this page for a list of all contributions to it.