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 $ \mathfrak N $ be a category with coproducts. An object $ U \in \mathop{\rm Ob} \mathfrak N $ is called small if for any morphism

$$ \phi : U \rightarrow \ \sum _ {i \in I } U _ {i} $$

where $ U _ {i} = U $, $ i \in I $, and $ \sigma _ {i} $ is the imbedding of the $ i $- th summand in the coproduct, there is a finite subset of the indices $ 1 \dots n $ such that $ \phi $ factors through the morphism

$$ U _ {1} + \dots + U _ {n} \rightarrow \sum _ {i \in I } U _ {i} $$

induced by $ \sigma _ {1} \dots \sigma _ {n} $. Sometimes a stronger definition is given in which it is not assumed that all summands in the coproduct $ \sum _ {i \in I } U _ {i} $ coincide with $ U $.

In varieties of finitary universal algebras the following conditions on an algebra $ A $ are equivalent: a) $ A $ is a small object of the category; b) $ A $ has a finite number of generators; and c) the covariant hom-functor $ H _ {A} ( X) = H ( A, X) $ 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 $ U $ is small if and only if the Abelian-group-valued functor $ H( U , -) $ 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 $ U $ to a coproduct should factor through a unique summand. In practice the term is rarely used outside the context of additive categories.