Categories meet semigroups in various ways
Bertalan Pécsi
TL;DR
The paper develops a categorical framework linking semigroups and category theory through wired categories and Karoubi envelopes, establishing an adjunction between semigroups and categories via a factoring through wired categories. It characterizes regular semigroups in terms of split epi–split mono factorizations of Karoubi envelopes and provides a factorization-based criterion for regularity. It introduces lax morphisms (lax spiders) and semigroupads, showing how Kleisli liftings correspond to semigroupad structures and how semigroup operations extend to $S^T$ for any semigroup $S$. It further analyzes how identities and central elements lift under semigroupad constructions, enabling systematic transfer of semigroup structure along endofunctors and highlighting the interaction between algebraic and categorical structures with potential applications in generalized monadic settings.
Abstract
This paper touches on several interaction points of semigroups and constructions from category theory: An adjunction is established between categories with selected arrows and semigroups. Regular semigroups are characterized by split epi - split mono factorization of the Karoubi envelope. We investigate how semigroupads (monads without requirement of unit transformation) map semigroups to semigroups and ensure certain properties provided they hold on meta level.