In case anyone is interested in delving deeper, I wrote a detailed series of posts on persistent homology including the fundamental math and getting a working python example: http://outlace.com/TDApart1.html
Homology (and Cohomology) are one of the key methodical innovations in pure Mathematics in the 20th century. It started as the topological concept described in the article, but the core idas apply much more general.
Todays: Algebraic Geometry, Number Theory, Category Theory, Singularity Theory, String Theory, etc. are full of "cohomology". There are thousands of different flavours in use, that measure different properties of "spaces" of interest.
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?
this is a pretty discussion but not a very good one if you don't already know some things.
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 π" will be denoted πΆπ(π)
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 βπβker βπ-1
instead of
>im βπ+1βker βπ
to stay with the notation in the previous sentence
A curve is a 1-dimensional space. Even though you draw a picture of it in a 2 (or maybe higher?) dimensional space, you should think of the set of points in the curve as being the only points that we care about.
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.
A closed curve could be the boundary of something, but it has no boundary itself. The boundary of an open curve is its endpoints, a closed curve has no endpoints.
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 one-dimensional shape, a boundary point is one you can stand on and move in one direction, but not in the other. So for an open line segment 0 < x < 1, there is no boundary, since at every point you can move in both directions. For a half-open line segment 0 < x <= 1 the point x=1 is the boundary, since you can move left but not right.
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 boundary of a line segment is its end points. Intuitively, it a line doesn't have end points, then it's a loop i.e. a closed curve without boundary
I've definitely been deep into some topology proofs where I suddenly proved a true fact couldn't possibly be true and then I realized I'd forgotten that not open doesn't imply not closed or something. It's funny the tricks that plays on your brain even when you know better.
That's what open and curve mean in topology. For example, the set of x with 0 <= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously confusing terminology, since sets can be both closed and open, or neither...
>That's what open and curve mean in topology. For example, the set of x with 0 <= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously confusing terminology, since sets can be both closed and open, or neither...
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.
It turns out that that's a completely different meaning of "homology".
"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.
This sort of homology[1] or this sort[2]? It looks like this is about the first one and proteins usually involve the second one. If they are more related than I expect, I would love to find out.
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.
