n-group (category theory)

In category theory, an n-group is a generalization of the notion of a group, specifically a groupoid enriched in (n-1)-groupoids. This provides a higher-categorical analogue of a group, capturing higher-order algebraic structures and homotopical information.

More formally, an n-group can be defined recursively:

• A 0-group is a set.
• An (n+1)-group is a groupoid enriched in n-groups.

Unpacking this enrichment, an (n+1)-group consists of:

• A set of objects.
• For any two objects, a collection of morphisms between them, which form an n-group.
• A composition law for morphisms which is suitably associative and unital, and respects the n-group structure of the morphism collections.
• Invertibility conditions relating to the composition.

In particular:

• A 1-group is precisely a groupoid. The morphisms between any two objects form a group.
• A 2-group is a category internally to the category of groups. Its objects form a set and the morphisms between them form a group. Composition and the group operation must be compatible. Equivalently, it is a strict 2-category with a single object where all morphisms are invertible (strictly).

n-groups are used to study higher homotopy theory and higher gauge theories. They provide an algebraic model for certain types of homotopy n-types. For example, the fundamental n-groupoid of a topological space can be thought of as a weak n-group.

They are also related to crossed modules, braided crossed modules and other higher algebraic structures. The study of n-groups is closely linked to that of higher categories and higher groupoids.