What is...homology categorifying?

Goal.
Explaining basic concepts of algebraic topology in an intuitive way.

This time.
What is...homology categorifying? Or: Modules, polynomials and numbers.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.

Slides.
http://www.dtubbenhauer.com/youtube.html

Website with exercises.
http://www.dtubbenhauer.com/lecture-a...

Singular homology.
https://en.wikipedia.org/wiki/Singula...

Hilbert-Poincaré polynomial.
https://en.wikipedia.org/wiki/Hilbert...

Euler characteristic.
https://en.wikipedia.org/wiki/Euler_c...

Categorification.
https://mathoverflow.net/questions/48...

Lefschetz and Brouwer.
https://en.wikipedia.org/wiki/Lefsche...
https://en.wikipedia.org/wiki/Brouwer...

Pictures used.
https://en.wikipedia.org/wiki/Sphere#...
https://simple.wikipedia.org/wiki/Cir...
https://en.wikipedia.org/wiki/Klein_b...
https://mathimages.swarthmore.edu/ind...

Hatcher’s book (I sometimes steal some pictures from there).
https://pi.math.cornell.edu/~hatcher/...

Always useful.
https://en.wikipedia.org/wiki/Counter...

Mathematica.
https://demonstrations.wolfram.com/Br...

#algebraictopology
#topology
#mathematics