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Category Theory

Table of Contents

1. "Overview"

I'm not going to pretend to talk about why category theory is useful. I've written such notes before, it'd be a distraction to re-hash them now. (It gives us a template to describe mathematical gadgets, basically. This template itself is a mathematical gadget, so it can be used on itself.)

Category theory exposes the magic underpinning relations like Monstrous Moonshine, Galois theory, or the Langlands programme (and many many more).

2. Categorification

Categorification is the process of finding category theoretic counterparts to classical mathematics. It's defined by decategorification reproducing a given mathematical gadget.

A seemingly dull example is how finite sets are categorified natural numbers. When we consider formal power series whose coefficients are particular natural numbers, the categorified counterpart is Feynman diagrams.

This can be related to the process of internalization.

3. References

3.1. Examples of Categorification

  • John Baez and Michael Shulman,
    Lectures on n-Categories and Cohomology.
  • Mikhail Khovanov,
    "Heisenberg algebra and a graphical calculus."
    Eprint: arXiv:1009.3295 [math.RT], 45 pages.
  • Alexander Kirillov Jr,
    "On the modular functor associated with a finite group."
    Eprint: arXiv:math/0310087 [math.QA], 7 pages.
  • Jacob Lurie,
    "Categorification of Fourier Theory".
    Youtube lecture, 2015
  • Ingo Runkel, Jurgen Fuchs, Christoph Schweigert,
    "Categorification and correlation functions in conformal field theory."
    Eprint: arXiv:math/0602079 [math.CT], 16 pages. International Congress of Mathematicians. Vol. III, pages 443–458, Eur. Math. Soc., Zürich, 2006.

3.1.1. 2-Vector Spaces

  • John C. Baez and Alissa S. Crans,
    "Higher-Dimensional Algebra VI: Lie 2-Algebras".
    TAC, 48 pages
  • Jeffrey C. Morton,
    "2-Vector Spaces and Groupoids".
    arXiv:0810.2361, 44 pages.

3.1.2. Galois Theory

  • Theo Johnson-Freyd,
    "Spin, statistics, orientations, unitarity".
    arXiv:1507.06297, offers a categorification of Galois theory. (See also handout)
  • Deligne's "Categories Tensorielles"
    English translation also discusses categorified Galois theory.

3.1.3. Number Theory and Stuff

  • Quoc P. Ho, Penghui Li, "Eisenstein series via factorization homology of Hecke categories". arXiv:2103.10137

3.1.4. Polynomials

  • Syu Kato, Anton Khoroshkin, Ievgen Makedonskyi, "Categorification of DAHA and Macdonald polynomials". arXiv:2103.10009, 37 pages
  • Mikhail Khovanov, Radmila Sazdanovic, "Categorification of the polynomial ring." Eprint: arXiv:1101.0293 [math.QA], 29 pages.

3.2. Higher Category Theory

Last Updated 2021-06-01 Tue 10:00.