finitely generated
Such that the functor operatorname Hom_C preserves those filtered colimits of…In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators: Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
Meaning
5 sensesIn any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
(not comparable)Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
(not comparable)Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set.
(abstract, not comparable, usually)Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)
(not comparable)Finitely generated as a (left) module over R.
(not comparable)Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).
Origin
From the study of generators. The motivation for calling topologies satisfying sense 1.5 "From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.
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