Inductive limit

A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathematics. An important special case of an inductive limit is the inductive limit of a directed family of mathematical structures of the same type. Let $ C $ be a directed pre-ordered set, that is, a reflexive transitive relation $ \prec $ is defined on $ C $ and for any two elements $ \alpha , \beta \in C $ there exists an element $ \gamma \in C $ such that $ \alpha \prec \gamma $ and $ \beta \prec \gamma $. Suppose further that a structure $ A _ \alpha $ is associated with each $ \alpha \in C $( for definiteness, suppose that the $ A _ \alpha $ are groups) and that for each $ \alpha \prec \beta $ homomorphisms $ \phi _ {\alpha \beta } : A _ \alpha \rightarrow A _ \beta $ are given satisfying the two conditions: $ \phi _ {\alpha \alpha } = 1 _ {A _ \alpha } $ for any $ \alpha \in C $ and $ \phi _ {\alpha \beta } \phi _ {\beta \gamma } = \phi _ {\alpha \gamma } $ for any $ \alpha \prec \beta \prec \gamma $ in $ C $. An equivalence relation $ \sim $ is introduced on the set $ \overline{A}\; = \cup _ {\alpha \in C } A _ \alpha $: The element $ x \in A _ \alpha $ is equivalent to $ y \in A _ \beta $ if $ x \phi _ {\alpha \gamma } = y \phi _ {\beta \gamma } $ for some $ \gamma $. The quotient set $ A = \overline{A}\; / \sim $ can then be endowed with a group structure: If $ x \in A _ \alpha $, $ y \in A _ \beta $ and $ \alpha \prec \gamma $, $ \beta \prec \gamma $, then the product of the equivalence classes represented by $ x $ and $ y $ is defined to be the equivalence class with representative $ ( x \phi _ {\alpha \gamma } ) ( y \phi _ {\beta \gamma } ) $. The resulting group $ A $ is called the inductive limit of the family of groups $ A _ \alpha $. There exists for each $ \alpha \in C $ a natural homomorphism $ \phi _ \alpha : A _ \alpha \rightarrow A $ associating to an element $ x \in A _ \alpha $ its equivalence class. The group $ A $ together with its homomorphisms $ \phi _ \alpha $ has the following property: For any system of homomorphisms $ \psi _ \alpha : A _ \alpha \rightarrow B $, $ \alpha \in C $, for which $ \psi _ \alpha = \phi _ {\alpha \beta } \psi _ \beta $ for $ \alpha \prec \beta $, there exists a unique homomorphism $ \psi : A \rightarrow B $ such that $ \psi _ \alpha = \phi _ \alpha \psi $ for any $ \alpha \in C $.

A generalization of the above construction of an inductive limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object $ A $ of a category $ \mathfrak K $ is called an inductive limit of the covariant functor $ F : \mathfrak D \rightarrow \mathfrak K $ if:

1) there exist morphisms $ \phi _ {D} : F ( D) \rightarrow A $, where $ D \in \mathop{\rm Ob} \mathfrak D $, such that $ F ( \alpha ) \phi _ {D _ {1} } = \phi _ {D} $ for any morphism $ \alpha : D \rightarrow D _ {1} $ in $ \mathfrak D $; and

2) for any family of morphisms $ \psi _ {D} : F ( D) \rightarrow B $, where $ D \in \mathop{\rm Ob} \mathfrak D $, such that $ F ( \alpha ) \psi _ {D _ {1} } = \psi _ {D} $ for any $ \alpha : D \rightarrow D _ {1} $ in $ \mathfrak D $, there exists a unique morphism $ \gamma : A \rightarrow B $ such that $ \psi _ {D} = \phi _ {D} \gamma $, $ D \in \mathop{\rm Ob} \mathfrak D $.

An inductive limit is denoted by $ ( A , \phi _ {D} ) = \lim\limits _ \rightarrow F $ or $ A = \lim\limits _ \rightarrow F $ or $ A = \lim\limits _ \rightarrow F ( D ) $. An inductive limit of a contravariant functor $ F : \mathfrak D \rightarrow \mathfrak K $ is defined as an inductive limit of the covariant functor $ F ^ { * } $ from the dual category $ \mathfrak D ^ {*} $ into the category $ \mathfrak K $.

Every pre-ordered set $ C $ can be regarded as a category whose objects are the elements of $ C $ and whose morphisms are all pairs $ ( \alpha , \beta ) $ for which $ \alpha , \beta \in C $ and $ \alpha \prec \beta $, with the obvious law of composition. In an arbitrary category $ \mathfrak K $, a family of objects $ A _ \alpha $, $ \alpha \in C $, and morphisms $ \phi _ {\alpha \beta } : A _ \alpha \rightarrow A _ \beta $, where $ \alpha \prec \beta $, can be regarded as the image of a functor $ F : C \rightarrow \mathfrak K $ if $ \phi _ {\alpha \alpha } = 1 $ and $ \phi _ {\alpha \beta } \phi _ {\beta \gamma } = \phi _ {\alpha \gamma } $ for $ \alpha \prec \beta \prec \gamma $. If $ \mathfrak K $ is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor $ F : C \rightarrow \mathfrak K $ coincides with the construction of the inductive limit defined above.

If $ \mathfrak D $ is a small discrete category, then an inductive limit of any functor $ F $ from $ \mathfrak D $ into an arbitrary category $ \mathfrak K $ is a coproduct of the objects $ F ( D) $, $ D \in \mathfrak D $. In particular, if $ \mathfrak D $ is empty, then an inductive limit is a left null or an initial object of $ \mathfrak K $. Cokernels of pairs of morphisms of any category $ \mathfrak K $ are inductive limits of functors defined on the category with two objects $ X $ and $ Y $ and four morphisms $ 1 _ {X} , 1 _ {Y} $ and $ \alpha , \beta : X \rightarrow Y $.

Every covariant functor $ F $ from an arbitrary small category $ \mathfrak D $ into a category $ \mathfrak K $ has an inductive limit if and only if $ \mathfrak K $ has coproducts and cokernels of pairs of morphisms.

References

[1]	P.S. Aleksandrov, "Topological duality theorems" Trudy Mat. Inst. Steklov. , 48 (1955) pp. Chapt. 1 (In Russian)
[2]	I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[3]	M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)

Comments

In the main article above, $ x \phi _ {\alpha \gamma } $ stands for $ \phi _ {\alpha \gamma } ( x) $: the value of the homomorphism $ \phi _ {\alpha \gamma } $ on the element $ x $.

Similarly, composition of homomorphisms is written backwards: $ \phi _ {\alpha \beta } \psi _ \beta $ means first apply $ \phi _ {\alpha \beta } $ and then $ \psi _ \beta $.

In English, the term "inductive limit" is usually restricted to limits over directed pre-ordered sets, the more general categorical concept being called a colimit. "Cokernels of pairs of morphisms" are commonly called coequalizers.

Dual to the notion of an inductive limit is that of a projective limit, also called an inverse limit.

References

[a1]	S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7