This repository contains the LaTeX source code for notes and materials prepared for a reading lecture on the Category of Diffeological Spaces. The lecture explores diffeological spaces as a generalized framework for smooth geometry and their categorical properties, particularly their interpretation as concrete sheaves on a concrete site.
Diffeological spaces generalize smooth manifolds, providing a robust framework for studying "generalized spaces." A diffeological space consists of a set ( X ) equipped with a collection of "plots"—smooth maps from open subsets of Euclidean spaces into ( X )—that satisfy three basic axioms. While individual diffeological spaces may exhibit geometric pathologies absent in smooth manifolds, the category ( \mathbf{Diff} ) of all diffeological spaces possesses numerous desirable categorical properties that the category of smooth manifolds lacks.
The notes include a concise introduction to diffeological spaces, the concept of a concrete site, and concrete sheaves. They demonstrate that ( \mathbf{Diff} ) can be identified as a category of "concrete sheaves on a concrete site," making it an archetypal example of a "generalized space." The notes also detail structural properties of this class of categories, emphasizing that any category of concrete sheaves on a concrete site—including ( \mathbf{Diff} )—is a quasitopos, with all limits and colimits.
The current version, demo.pdf, is missing some diagrams and proofs.
notes/: Contains detailed LaTeX files for the lecture notes.diagrams/: Folder for diagrams and figures used in the notes.bibliography.bib: References used in the notes.main.tex: The main LaTeX file to compile the entire document.
To fully appreciate the material, familiarity with the following topics is recommended:
- Basic definitions: Category, natural transformations
- Limits and colimits
Contributions and feedback are welcome! If you have suggestions for improving the notes, please open an issue or submit a pull request.
This project is licensed under the MIT License. See the LICENSE file for details.