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Kan extendable subcategories and fibrewise topology

Moncef Ghazel

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.

Kan extendable subcategories and fibrewise 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.

Paper Structure

This paper contains 17 sections, 105 theorems, 96 equations.

Table of Contents

  1. Reflective subcategories
  2. Idempotent codensity monads
  3. Left and right Kan extendable subcategories and their properties
  4. Compactifications
  5. Reflective subcategories of $\mathsf{Top}_B$
  6. Fibrewise compact spaces
  7. Fibrewise weak and $k$-Hausdorfifications
  8. Left Kan extendable subcategories of $\mathsf{Top}_B$
  9. Cartesian closed category of $\mathcal{W}$-generated objects
  10. A fibrewise Day's theorem
  11. The category $\mathsf{kTop}_B$ of fibrewise compactly generated spaces
  12. Cartesian closed subcategories of $\mathsf{kTop}_B$ and $\mathsf{kTop}$
  13. Fibrewise sequential spaces
  14. Fibrewise Alexandroff spaces
  15. Limits in a slice category
  16. ...and 2 more sections

Key Result

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Theorems & Definitions (220)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • ...and 210 more