Category of representations

In representation theory, the category of representations of some algebraic structure A has the representations of A as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects (see Maschke's theorem for the case of finite groups).

The Tannakian formalism gives conditions under which a group G may be recovered from the category of representations of it together with the forgetful functor to the category of vector spaces.

The Grothendieck ring of the category of finite-dimensional representations of a group G is called the representation ring of G.

Depending on the types of the representations one wants to consider, it is typical to use slightly different definitions.

For a finite group G and a field F, the category of representations of G over F has

The category is denoted by ${\displaystyle \operatorname {Rep} _{F}(G)}$ or ${\displaystyle \operatorname {Rep} (G)}$.

...
Wikipedia