Kan extendable subcategories and fibrewise topology

TL;DR

This paper employs pointwise Kan extensions to generate and analyze new subcategories, focusing on fibrewise topological spaces and establishing conditions for their cartesian closedness.

Contribution

It introduces a general method using Kan extensions to create and study subcategories, with specific applications to fibrewise topological spaces.

Findings

Fibrewise compactly generated spaces are cartesian closed under certain separation axioms.

Fibrewise sequential and Alexandroff spaces are also cartesian closed with appropriate base space conditions.

The methods are broadly applicable to category theory and topology.

Abstract

We use pointwise Kan extensions to generate new subcategories out of old ones. We investigate the properties of these newly produced categories and give sufficient conditions for their cartesian closedness to hold. Our methods are of general use. Here we apply them particularly to the study of the properties of certain categories of fibrewise topological spaces. In particular, we prove that the categories of fibrewise compactly generated spaces, fibrewise sequential spaces and fibrewise Alexandroff spaces are cartesian closed provided that the base space satisfies the right separation axiom.