In mathematics, a direct limit of groups is the direct limit of a direct system of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called finitary or stable groups, though this latter term normally means something quite different in model theory.
Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit. This is a priori surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.
The notion of a direct limit captures many vague but intuitive ideas of "group limits": the finite Symmetric groups should limit to an infinite symmetric group and the subgroups of a group should limit to , in some sense. Under the direct limit construction, group families (symmetric groups, dihedral groups, general linear groups , etc) will generally limit to the finitary or stable subgroup of the corresponding infinite group: the groups don't limit to the permutation group of a countable set, , but do limit to its subgroup of permutations which permute only finitely many objects. We'll also often see that recovering a group as a direct limit of its subgroups can be done simply (and sometimes only) with its finitely generated subgroups. Direct limits have a more general definition in Category theory, which reduces to the definition below in the category of groups, and more generally, any concrete category.
Let be a set with a transitive, reflexive binary relation (a preorder). We call a directed set if, for all and in , there exists some such that and . Let be a family of groups indexed by with group homomorphisms for all in such that
The pair is called a direct system, and we form the set. The direct limit of the direct system is denoted by and is defined on equivalence classes of the disjoint union of the with for and if , where is the upper bound of and . That is,
For , , and upper bound of and , we define the binary operation on by setting , where the multiplication is performed in . The operation is well defined by the compatibility condition on the , and associativity follows from associativity in the . Since each map is a homomorphism, all identities lie in the same equivalence class, and this class forms the identity of . Finally, the inverse of for is simply .
Like many categorical constructions, direct limits are unique in a strong sense: for two direct limits and of a direct system, there exists a unique isomorphism .