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1 Formal definition

1.1 Algebraic objects

We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let {A_i | i ∈ I} be a family of groups indexed by I and suppose we have a family of homomorphisms f_ij : A_j → A_i for all i ≤ j (note the order) with the following properties:

1. f_ii(x) = x for all x ∈ A_i,
2. f_ik = f_ij O f_jk for all i ≤ j ≤ k.

Then the pair (A_i, f_ij) is called an inverse system of groups and morphisms over I.

We define the inverse limit of the inverse system (A_i, f_ij) is defined as a particular subgroup of the direct product of the A_i's:

The inverse limit, A, comes equipped with natural projections π_i : A → A_i which pick out the ith component of the direct product. The inverse limit and the natural projections satisfy a universal property described in the next section.

This same construction may be carried out if the A_i's are sets, rings, modules (over a fixed ring), algebras (over a fixed field), etc., and the homomorphisms are homomorphisms in the corresponding categoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan. The inverse limit will also belong to that category.

1.2 General definition

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (X_i, f_ij) be an inverse system of objects and morphismIn mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are gros in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms π_i : X → X_i (called projections) satisfying π_i = f_ij O π_j . The pair (X, π_i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u : Y → X making all the "obvious" identites true; i.e. the diagram

must commuteHomological algebra Category theory In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtai for all i, j. The inverse limit is often denoted

with the inverse system (X_i, f_ij) being understood.

Unlike for algebraic objects, the inverse limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any another inverse limit X′ there exists is a unique isomorphismIn mathematics, an isomorphism is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition: :The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part X′ → X commuting with the projection maps.

We note that an inverse system in category C admits an alternative desription in terms of functorFor the usage in computer science, see the function object article. In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all ( small) categories. Functors were first conss. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j iff i ≤ j. An inverse system is then just a contravariant functor I → C.