Some remarks on multicategories and additive categories

TL;DR

This paper explores the relationship between categories, multicategories, and additive categories, highlighting coreflective embeddings and adjunctions that connect algebraic theories and module representations.

Contribution

It establishes coreflective embeddings of categories into multicategories and lifts these to the cartesian level, linking module theories and algebraic structures.

Findings

Categories embed into multicategories via discrete cocone construction

The adjunction between categories and multicategories lifts to preadditive and cartesian multicategories

A direct connection between module theories over rigs and algebraic models is demonstrated

Abstract

Categories are coreflectively embedded in multicategories via the "discrete cocone" construction, the right adjoint being given by the monoid construction. Furthermore, the adjunction lifts to the "cartesian level": preadditive categories are coreflectively embedded (as theories for many-sorted modules) in cartesian multicategories (general algebraic theories). In particular, one gets a direct link between two ways of considering modules over a rig, namely as additive functors valued in commutative monoids or as models of the theory generated by the rig itself.