Intuition for monoidal categories: Every (small) monoidal category may also be viewed as a “categorification” of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category’s objects and whose binary operation is given by the category’s tensor product.
Braided monoidal categories: natural isomorphisms satisfying the hexagon axiom. symmetric monoidal categories: . Example: product of two sets, tensor product of vector spaces, graded vector space with braiding given by Koszul sign convention
An algebra in a monoidal category is a monoidal object (bilinearity is distributive law). Similarly we can define coalgebras. If is braided and are algebras, then so is . A bialgebra in a braided category is a algebra and coalgebra s.t. comultiplication is a map of algebras. It is moreover a Hopf algebra if there is an antipode (like inverse) such that is equal to the composite .
Monoidal functor (lax): is monoidal if there exists natural transformation (colax if it is the other direction)
If is an algebra in and is lax monidal then is an algebra in (coalgebra if it is colax monoidal)
A very nontrivial class of braided monoidal cateogies is that of Yetter-Drinfeld modules: group, be the category with object right -modules which decompose such that where and morphisms are linear maps preserving the action and the grading. The monoidal structure is with action via diagonal action. The braiding is going to send to (similar to semidirect product) where .