It’s impossible to get far in reading 20th-century mathematics without encountering the word cohomology. Cohomology & schemes are the subject of Hartshorne’s classic, where you can find out (Appendix C) that the Weil conjectures were resolved by defining a thing called l-adic cohomology.
Schemes are like varieties = cycles. And cohomology is a way (ok, apparently various ways!) of calculating shape.
So what does it mean to “define” “a” cohomology “theory”? What does it mean that Dror Bar-Natan is fascinated by Khovanov homology? That Dale Husemöller wants to interpolate beween different cohomology “theories”—crystalline, étale, Hodge, and so on?
Mathematicians drop the word “theory” like rappers drop the “N” bomb.
One starts with simplicial homology
This “theory” takes triangular decompositions of a space into triangles and returns a chain complex with a (everywhere ∂²=0) boundary map.
Why do chain complexes come up in this topic? To algebraicise the geometric idea here, you set up maps from the higher-dimensional things to lower-dimensional things. (In case of simplicial homology, the “things” are triangles and pyramids. In keeping with Eilenberg & Mac Lane’s fundamental rule of homology, ∂² needs to always zero out.
(We are doing this in a “formal” sense—meaning that an entry might be like that blue (solid) pyramid over there., for example in the sentence 3×that blue (solid) pyramid over there (hey, it came from the (3,5) position of the filtered complex− that blue (solid) pyramid over there − 5
(Hey — you already knew that mathematical sentences get too long — just like you knew that topological (not algebraic) functions can get so wiggly that it’s not worth trying to explicitly describe them.
The fact that these homology sentences can be so horrid, and yet the cancellation criterion ∂²=0 is so simple, is–I think–why mathematicians regard the Eilenberg-MacLane condition as “deep”.
According to Matt H, the boundary map is what
At some point people figured out that you can cover the possible topological types …………… and thus be able to say something about X, again without having to go into painfully boring detail. —- For applications people, this may mean that the things we want to talk about fall ………… or it may not.
(If the space is X you will see people write BG(X) – the letter B here means Eilenberg-MacLane topological type.