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