# 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.

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.