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
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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
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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.
Small object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_object&oldid=48736