Group Actions can be seen as functors

up:: Functor

Given a Group Action $\alpha :G\times X\to X$ from a Group $G$ onto a set $X$, one can see it as a functor between the associated Single-Element Category $BG$ and the Set Category $Set$. In fact, this allows for a generalization of group actions to any type of category.

This point of view leverages the fact that Groups can be seen as automorphisms in single-element categories, from which one sees each group element as a symmetry transformation.

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References

• What is a Functor? Definition and Examples, Part 1