Overcategories and free monoids for overcategories

TL;DR

This paper extends foundational theorems and constructions like free monoids to the setting of overcategories, facilitating the development of higher category theory.

Contribution

It generalizes key categorical theorems and free monoid constructions to overcategories, building on prior work to support higher category development.

Findings

Freyd's adjoint theorem holds in overcategories

Barr and Wells' theorem applies to overcategories

Free monoid construction is valid in overcategories

Abstract

An overcategory with base category C is merely any functor into C. In this paper we extend the work of Dominique Bourn and Jacques Penon ("Cat\'egorification de structures d\'efinies par monade cart\'esienne") on overcategories. In particular we show that Freyd's adjoint theorem, a theorem of Barr and Wells ("Toposes, Triples and Theories"), all remain true in the context of overcategories. We also show that a free monoid construction remains valid in the context of overcategories. The motivation for this study is the development of higher categories as found in the work of Dominique Bourn and Jacques Penon ("Cat\'egorification de structures d\'efinies par monade cart\'esienne").