Small object

A concept which singles out objects in a category that have intrinsically the properties of a mathematical structure with a finite number of generators (finite-dimensional linear space, finitely-generated group, etc.). Let be a category with coproducts. An object is called small if for any morphism

where , , and is the imbedding of the -th summand in the coproduct, there is a finite subset of the indices such that factors through the morphism

induced by . Sometimes a stronger definition is given in which it is not assumed that all summands in the coproduct coincide with .

In varieties of finitary universal algebras the following conditions on an algebra are equivalent: a) is a small object of the category; b) has a finite number of generators; and c) the covariant hom-functor commutes with colimits (direct limits) of directed families of monomorphisms. Property c) is often taken as the definition of a finitely-generated object of an arbitrary category.

Comments

In an additive category, an object is small if and only if the Abelian-group-valued functor preserves coproducts. Some authors take this as the definition of smallness in non-additive categories: it produces a more restrictive condition than the one above, equivalent to the requirement that every morphism from to a coproduct should factor through a unique summand. In practice the term is rarely used outside the context of additive categories.