Groups can be seen as automorphisms in single-element categories

up:: Category

To every Group $G$, one can associate a Single-Element Category $BG$, whose morphisms are all elements in $G$. Due to the algebraic structure of $G$, these morphisms will all be automorphisms (endomorphic & invertible).

This way, it is straightforward to say that group elements are really symmetry maps: to each element $g\in G$, there is a morphism which preserves this object’s structure ─ such as rotations of $\pi$ for a line ($2$-gon), rotations of $\frac{2\pi }{3}$ for triangles, and so on.

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References

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