Group

A group is a set with an associative multiplication, a unit , and an inverse for every element. Equivalently, a group is a category with a single object in which every morphism is invertible. The morphisms are the group elements, composition is multiplication, and the identity morphism is . Dropping invertibility leaves a monoid, the one-object category with no further condition.

Example. Actions and representations are functors

A functor out of the one-object category names where the object goes and sends each element to an endomorphism of it. Into it is a group action, a copresheaf , under which the object becomes a set and each an automorphism of . Into it is a linear representation , under which the object becomes a vector space and each an invertible linear map.

Proposition. An equivariant map is a natural transformation

A map of -actions is a natural transformation exactly when it is -equivariant. Given two actions on and , equivariance means for all , which is precisely the naturality condition on the single component . The naturality square is the equivariance square. -invariance is the case where is trivial, so is constant on orbits.

References

• Mac Lane, Categories for the Working Mathematician, 2nd ed. (Springer, 1998)