Direct limit of groups

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 stable groups, though this 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.

Examples

Each family of classical groups forms a direct system, via inclusion of matrices in the upper left corner, such as ${\displaystyle \operatorname {GL} (n,A)\to \operatorname {GL} (n+1,A)}$. The stable groups are denoted ${\displaystyle \operatorname {GL} (A)}$ or ${\displaystyle \operatorname {GL} (\infty ,A)}$.

Bott periodicity computes the homotopy of the stable unitary group and stable orthogonal group.

The Whitehead group of a ring (the first K-group) can be defined in terms of ${\displaystyle \operatorname {GL} (A)}$.

Stable homotopy groups of spheres are the stable groups associated with the suspension functor.