Editing Hopf algebra (section)

==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]]".<ref>[http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra/ Group = Hopf algebra « Secret Blogging Seminar<!-- Bot generated title -->], [https://www.youtube.com/watch?v=p3kkm5dYH-w Group objects and Hopf algebras], video of Simon Willerton.</ref>