Abstract

The aim of this talk is to illustrate one of MacLane’s slogans in his book ‘Categories for the Working Mathematician’, namely the centrality of the concept of a monoid in mathematics.

The talk will consist of two parts. In the first part I will explain the basic definitions beginning with internal monoids and internal groups in categories with finite products and generalize them to the framework of (symmetric) monoidal categories, i.e., to internal algebras and internal Hopf algebras. In course of the generalization we will understand the emergence of comonoids (coalgebras) and bimonoids (bialgebras). In the second part I will sketch a hierarchy of algebraic structures with their respective tensor products applying the concepts and elementary observations from the first part to the category SETS . We will thereby encounter with familiar structures like groups, rings and algebras (over commutative rings) as well as less familiar structures like monoids, rigs, and algebras over commutative rigs. Finally, I will turn to certain adjunctions in this hierarchy and use them to explain some of the standard (and universal) examples of Hopf algebras.