Analogy With Groups
Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module. In this case:
- the field K is replaced by the 1-point set
- there is a natural counit (map to 1 point)
- there is a natural comultiplication (the diagonal map)
- the unit is the identity element of the group
- the multiplication is the multiplication in the group
- the antipode is the inverse
In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element".
Read more about this topic: Hopf Algebra
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