Generator of a category

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generating object

An object in a category $ \mathfrak C $ such that the corresponding representable functor $ \mathfrak C \rightarrow \mathop{\rm Set} $" detects differences between objects of the category" in a suitable sense. There are two precise formulations of this concept in common use: for the first, an object $ G $ is said to be a generator (sometimes strong generator or proper generator) if, given a non-invertible monomorphism $ m: A ^ \prime \rightarrow A $ in $ \mathfrak C $, there exists an $ h: G \rightarrow A $ which does not factor through $ m $. For the second, $ G $ is said to be a generator if, given a pair of morphisms $ f, g: A {} size + 3 { {} _ \rightarrow ^ \rightarrow } B $ with $ f \neq g $, there exists an $ h: G \rightarrow A $ with $ fh \neq gh $; an object with this property is called a separator by some authors.

In any category with equalizers, the property of being a generator in the first sense implies that of being a generator in the second sense. The converse is true if the category is balanced (i.e. has the property that any morphism which is both monic and epic is an isomorphism) but not in general: for example, in the category of topological spaces the one-point space is a generator in the second sense but not in the first. In the category of sets, a singleton set (or indeed any non-empty set) is a generator in both senses; in a variety of universal algebras, the free algebra on any non-empty set is a generator in both senses.

A generalization of the notion of a generator is the notion of a generating set of objects or set of generators (also separating set, etc.). A set of objects $ \{ {G _ {i} } : {i \in I } \} $ is said to be a generating set (in the first or second sense) if it satisfies the appropriate condition above with "there exists an h: G A" replaced by "there exists an h:Gi A for some i I" . In an Abelian category (more generally, in any category with a zero object, cf. Null object of a category) with coproducts, the existence of a generating set implies the existence of a generator, since one can simply take the coproduct of the objects in the generating set, but this is not true in more general categories. For example, in the category of sheaves (of sets) on a topological space $ X $, the sheaves of sections of open subsets of $ X $ form a generating set (in both senses), but there is no single generator if $ X $ is non-trivial.

In a category with coproducts, the existence of a generating set (in the second sense) of projective objects (cf. Projective object of a category) implies that every object is an epimorphic image of a projective object (namely a suitable coproduct of copies of the generators). For this reason, the assumption that an Abelian category has a projective generator plays an important role in homological algebra.

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