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A concept which singles out objects in a [[Category|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|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.