Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.
These objects may be groups, rings, vector spaces or in general objects from any category.
The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects.
The direct limit of the objects
This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.
Direct limits are a special case of the concept of colimit in category theory.
We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used in any category.
In this section objects are understood to consist of underlying sets equipped with a given algebraic structure, such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc.
Its underlying set is the disjoint union of the
's modulo a certain equivalence relation
Intuitively, two elements in the disjoint union are equivalent if and only if they "eventually become equal" in the direct system.
An equivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its images under the maps of the direct system, i.e.
One obtains from this definition canonical functions
sending each element to its equivalence class.
The direct limit can be defined in an arbitrary category
be a direct system of objects and morphisms in
is a universally repelling target
Unlike for algebraic objects, not every direct system in an arbitrary category has a direct limit.
If it does, however, the direct limit is unique in a strong sense: given another direct limit X′ there exists a unique isomorphism X′ → X that commutes with the canonical morphisms.
Direct limits are linked to inverse limits via An important property is that taking direct limits in the category of modules is an exact functor.
This means that if you start with a directed system of short exact sequences
and form direct limits, you obtain a short exact sequence
We note that a direct system in a category
admits an alternative description in terms of functors.
A notion closely related to direct limits are the filtered colimits.
Here we start with a covariant functor
One can show that a category has all directed limits if and only if it has all filtered colimits, and a functor defined on such a category commutes with all direct limits if and only if it commutes with all filtered colimits.
(consider for example the category of finite sets, or the category of finitely generated abelian groups).
in which all direct limits exist; the objects of
The term "inductive limit" is ambiguous however, as some authors use it for the general concept of colimit.