Scalars, Monads, and Categories

Understand

This chapter describes interrelations between: (1) algebraic structure on sets of scalars, (2) properties of monads associated with such sets of scalars, and (3) structure in categories (esp.

• Lawvere theories) associated with these monads.
• These interrelations will be expressed in terms of "triangles of adjunctions", involving for instance various kinds of monoids (non-commutative, commutative, involutive) and semirings as scalars.
• It will be shown to which kind of monads and categories these algebraic structures correspond via adjunctions.