Homology
tungsteno.io
tungsteno.io
https://towardsdatascience.com/persistent-homology-with-exam...
Much of category theory was initially created to deal with "complexes of abelian objects" that are used to define cohomology theories.
Working with cohomology was my bread and butter, when working as a mathematician. I always hoped to apply sone idas to Computer Science as well, when I entered the field, but I did not find any sane ways to make use of that "technology". E.g. homology of graphs is pretty boring, since they are only 1dimensional. Is anyone aware of any CS applications of Homology?
for example
>the holes of dimension 0 are unconnected components
in this sentence
>the holes of dimension 0 are unconnected components, the holes of dimension 1 may be surrounded by a loop, whereas those of dimension 2 may be enclosed by surfaces
stands out because it's the only classification that's formal (i actually don't know what the meaning of unconnected component here is but i suspect it means a point?).
then C(X) is defined informally as
>The set of "objects of dimension 𝑘" will be denoted 𝐶𝑘(𝑋)
but this informal definition doesn't say explicitly that it's the set of objects in X (until later).
then orientation is defined in a very confusing way
>for now it will be some additional property that takes two opposite values ("one the negative of the other one")
but i will say that the discussion of quotienting was kind of an aha moment for me (about why you'd rather work in the quotient than the set itself) and i've been dipping toes into algebra for a loooooong time.
anyway you kind of have to read it like a poem (in that you should try to get a feeling rather than a precise idea) but that's not very useful.
also i'm interested in how the diagrams were created. looks like latex + tikz?
also it would've make more sense to show
>im ∂𝑘⊂ker ∂𝑘-1
instead of
>im ∂𝑘+1⊂ker ∂𝑘
to stay with the notation in the previous sentence
In a closed curve, you can never fall out of the space by moving around in it. In a curve with endpoints, you can fall out of the space by walking across one of the endpoints, so the endpoints are considered to be boundary components.
It's important to distinguish having a boundary vs. being the boundary of something, since in some sense the difference between the two is the whole point of homology.
For a two-dimensional shape (or more formally "manifold"), an interior point is one which locally looks like the plane (you can move in all directions), and a boundary point is one which locally looks like a half-plane. All the boundary points taken together will be a one-dimensional manifold _without boundary_, which is pretty neat. For example, take an annulus (a 2D shape like a CD-rom). The boundary of the annulus is two one-dimensional manifolds (the boundary circles), and those circles have no boundary.
The disk that it encloses, indeed, has a boundary (the curve in question).
You have it backwards; [0, 1] is closed and (0, 1) is open. This gives the terminology "open interval" and "closed interval".
The confusion about "why does a closed curve have no boundary" likely comes more from the word "boundary". The point is that for an n-dimensional object, the boundary is (n-1)-dimensional. For a curve, which is 1-dimensional, the boundary is 0-dimensional, ie points-- so we're looking for endpoints, and a closed curve doesn't have any.
But a closed curve doesn’t mean the same thing as a curve which is a closed set. A closed curve is a continuous image of S^1 (the circle) , yeah?
"Homology" really just means something like "correspondence". So it has, among its meanings,
1. the one in the article here: a way of associating topological spaces with algebraic structures, so that you can study one by studying the other.
2. the biological one: instances where two (parts or aspects of) living things are similar on account of common descent.
It's not 100% impossible that the two notions might come into contact with one another. DNA can become knotted, and there are enzymes called topoisomerases that enable DNA strands to pass through one another to unknot them. (Maybe something similar happens with proteins?) Knots are topological objects and homology is one of the tools mathematicians use to study them. Perhaps one day it will turn out that particular DNA or protein sequences tend to lead to particular sorts of knotting that are biologically significant (e.g., maybe they control the transcription rate of DNA or the shape of proteins), and perhaps it will turn out that they are selected for, and then perhaps from time to time biologists will find regions in two organisms' DNA or proteins that exhibit mathematically similar kinds of knotting because they're descended from an ancestor that found that kind of knotting useful. In that case they would have homology in their homology.
(Probably (k)not, though.)
You will need a strong grasp of category theory and topology in order to tackle this subject, though. The article serves as a good introduction, although it itself requires a mathematical background.