Monoidal internalization

Hopf monoid object

A Hopf monoid in a symmetric monoidal category is a bimonoid with the additional structure of an antipous,1 an endomorphism for which

c

holds. The antipous law states precisely that the antipous is the convolution-inverse of the identity map, whence such a map is unique when it exists. In a cartesian category the antipous law is equivalent to the inverse law of a group object, so that Hopf monoids are naturally viewed as a generalization of groups.

See also

#state/tidy | #lang/en | #SemBr

Footnotes

  1. Antipous (Gk. ἀντίπους) is the singular of the more common plural antipodes (Gk. ᾰ̓ντῐ́ποδες), although many authors use the back-formation antipode. ↩