Scalars, Monads, and Categories
Dion Coumans, Bart Jacobs
TL;DR
This chapter explores the deep connections between algebraic scalar structures, monads, and categorical frameworks like Lawvere theories, using adjunctions to relate various algebraic and categorical structures.
Contribution
It establishes a comprehensive framework linking scalar algebraic structures, monads, and categories through adjunctions, clarifying their interrelations.
Findings
Identifies how scalar algebraic structures correspond to specific monads.
Shows the categorical structures associated with different scalar monoids and semirings.
Uses adjunctions to relate algebraic and categorical properties.
Abstract
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.
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