Groups can be seen as automorphisms in single-element categories

up:: Category

To every Group , one can associate a Single-Element Category , whose morphisms are all elements in . Due to the algebraic structure of , 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 , there is a morphism which preserves this object’s structure ─ such as rotations of for a line (-gon), rotations of for triangles, and so on.

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References

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