Homology for Normal Humans
My friend asked me for help understanding this informationtheory paper. Specifically, what is homology?
I’ll walk you through the gradedchaincomplex of a 3simplex = triangular pyramid, but know that you could extend this kinda easily by sticking simplices together—that’s a “simplicial complex”. You could also stick nspheres onto that, and have a “CWcomplex”. I won’t get around to describing homology, but just knowing what a graded complex is, I think gets you close, and is easier to digest at one sitting. The overall message is that this definition of “shape” relies on looking at all the pieces together—“the pieces” being whole, one dimension lower, one dimension lower, one dimension lower, ….
Graded chain complexes
Starting from ground zero, I think the first thing anyone could learn about homology is the setting. Mathematicians tried probing spaces with homotopies but found the logic of homotopies to be hairy. They needed to invent homology, and in order to do that they needed to invent graded chain complexes.
Grading
So they invented the graded complex. Grading occurs in a couple settings; I drew the “simplicial” (triangular) setting above. Another setting is with polynomials: I learned at age 13 that x³+2x⁴
doesn’t combine to 3x⁷
.
† (Like Fractions, writing polynomial addition as algebra makes people want to do things that make no logical or geometric sense. People write ½ + ¼ = ⅙ or ½ + ½ = ¼, but would never write ◐ + ◑ = ◔.
‡ Another setting could be 1000 + 10
does not equal 2000
, but nobody makes this mistake.)
Grading does show up elsewhere but let’s leave off that.
Grading is not about stuff that’s totally incomparable. There are ways to combine x³
and x⁵
; addition just isn’t it. You can think of

x⁵
andx³
, or  ▲ and
boundary(
▲)
≝ △, or  △ and
boundary(
△)
≝ ∴ or  ■ and ∂(■) ≝ □
as being alike, kindof comparable, but of different dimension.
Complexes
This is the part I think anyone can understand. The idea of complexes, which are the setup to homology, is to consider the entirety of {pyramid, triangular faces, edges, vertices}
at once, or at least several of the neighbourly pieces at one time.
This is just a way of defining and setting up the system of inferences—a bit like defining classes in programming. There’s nothing that a tenyearold can’t conceive or understand in “Think about the faces, the edges, and the corners at once”. (The mental heavylifting is in working with the harder spaces, figuring out that this is a good setup, proving that it is, calculating homology, doing things with them, and relating this to other areas of maths.)
The word “complex” just means to think of {pyramid, ▲+▲+▲+▲, , ••••}
all together.
Chains of ∂ Maps
The → connection in simplicial chain complexes … → pyramid → ▲ → △ → ∴ → ∅ is the boundary map ∂. (each → being a homomorphism).