Kan Extensions in Context of Concreteness

TL;DR

This paper explores the theory of Kan extensions and their relation to concrete categories, introducing new concepts and strengthening existing theorems in category theory.

Contribution

It extends Kan extensions to cones, introduces l-algebraic categories, and weakens conditions in Beck's theorem using Kan extensions.

Findings

Limiting cones create Kan extensions

L-algebraic categories are natural and well-behaved

Weakening of Beck's theorem assumptions

Abstract

This paper contains results from two areas -- formal theory of Kan extensions and concrete categories. The contribution to the former topic is based on the extension of the concept of Kan extension to the cones and we prove that limiting cones create Kan extensions. The latter topic focuses on two significant families of concrete categories over an arbitrary category. Beck categories are defined by preservance properties while newly introduced l-algebraic categories are described by limits of categories of functor algebras. The latter family is shown to be rather natural. The well known Beck's theorem states that the monadic categories are precisely the Beck categories with free objects. We strengthen this theorem by weakening the assumptions of the existence of free objects and we replace it by existence of some Kan extensions, namely the pointwise codensity monads. Moreover, using the result on Kan extensions of cones we show that for l-algebraic categories even weaker assumption fits.