This is the question at the moment:

The original question was whether these two expressions are topologically equivalent, and I was was unaware that there was a theory which studies these symmetries.

Are the two expressions isomorphic by category theory, and how do they relate to topology? A more general question would be one related to Pascal's triangle, and its topology as we move to higher levels of abstraction.

If two things are symmetric, they are topologically equivalent.
nCk is symmetric to nCn-k,
β΄they are topologically equivalent. π²


Are nCk = nCn-k = 2 + 1 = 1 + 2, equivalent from the point of view of topology, because they are symmetric?

As different geometric objects are the same, namely, simple closed curves, these things are also the same, namely symmetric things

If things are topologically equivalent, are their abstractions topologically equivalent?

The geometric shape of a triangle is a single closed curve. Is the abstract representation of a triangle a single closed curve?

What are the basic basics to understand to explore/understand this question?

How to edit it further, for Math Stack Exchange, so that it is a good question for Math Stack Exchange?

• Did you think there was a topology defined on 6C2 or on 6C4? You need a topology defined to do "an elementary topological exercise". Commented May 10 at 20:29
• "I am at a loss at what makes this a good question for Math SE, as to me a mathematics question is just any maths related question one is curious about." Not every math-related question is a good question for Math.SE. "How can I use my proof of the Riemann Hypothesis to improve conditions for penguins in the Sahara?" would (I hope!) get closed and deleted quick smart. You may be familiar with math.meta.stackexchange.com/questions/9959/… β sometims, the only way to improve a question is to not ask it. Commented May 10 at 22:57
• @GerryMyerson We could make that into a good question! It could easily be a starting point of a fictional story, where penguins indeed inhabit the Sahara. You could make versions of this starting point and move beyond the cliche, and explore uses of your proof, and probably be able to reach many interesting ideas. Visualising abstract mathematical concepts vividly in stories is an excellent way to explore them, and to take them further. How does the Riemann Hypothesis contribute to your world, how does it visualise? Perhaps it shapes the landscape, or the penguins discuss creative ideas of it. Commented May 12 at 18:34
• Taking two seemingly unconnected ideas and stretching your mind to make connections, and take them further is not a new method. It is a fruitful way of creating stuff. I can imagine this question on Maths SE people contributing all sorts of ideas and developing a story, so that we can perhaps make the Riemann Hypothesis much more accessible, through storytelling. Commented May 12 at 18:36
• @AttilaVajda The fact that you would consider that a good question for this website indicates that you don't really understand the goal of Math SE. Speculation is not on-topic here. Fiction is not on-topic. Questions which do not have authoritative, objectively correct answers are (with some small exceptions) not on-topic here. That doesn't make these bad questions---it simply means that they don't fit on this website.
– Xander Henderson Mod
Commented May 12 at 18:43
• Which website do you recommend if one is interested in exploring the basic connections of the binomial coefficient, topology and category theory? You can have a million similar, burning questions when you are curious to study mathematics, and from the basicest exercises. You start a simple number theory exercise, and abstracting the numbers, you wonder what arrows are there, or what is the abstract algebra of it, or what open questions are there. Where to ask those questions? Commented May 12 at 18:49
• A more general, simpler form question, probably question in this case is how to find similarities, of nCk and nCn-k, what interesting, fascinating connections, visualisations exist? How to explore those patterns? These are mathematics questions, and mathematics questions often have no right or wrong answers. People are exploring these questions, and collaboration can make it easier to explore these things. Commented May 12 at 18:52
• @AttilaVajda I have no idea. My job is to moderate this site, not tell people where to go for dog grooming or speculative fiction or off-the-wall conjectures about the connections between category theory, topology, and binomial coefficients. I can really only tell you what is on- or off-topic here, not what might be on-topic elsewhere.
– Xander Henderson Mod
Commented May 12 at 18:53
• How to make this into a good research question, which is apparently a core component of doing mathematics, and a good indication of the kinds of questions that often lead to more profound questions in mathematics research, are the kinds of questions children have. Commented May 12 at 18:55
• "Where to ask those questions?" Start a blog, Attila; start a blog. There you will be free to ask questions about the topological effect of six-choose-two on Saharan penguins, or whatever else comes into your head. Commented May 12 at 22:33
• Okay! @GerryMyerson by the way, you can do it on Stack Exchange :) stackoverflow.blog/2012/05/22/encyclopedia-stack-exchange Also, ChatGPT is very good for generating stories, and ideas for accessible expressions for abstract concepts. Commented May 14 at 10:18
• Chatgpt is very good for generating crap. Commented May 14 at 11:54

The questions of yours that I have seen so far are indicative of someone who does not know what they are doing, but is reaching for all sorts of cool-sounding concepts and confusing things further. "Topological equivalence" is just a complete red herring when discussing $$\binom{n}{k}=\binom{n}{n-k}$$. If you want to keep using MSE, you need to get a bit more grounded - you need to be able to clearly articulate your questions so others can understand them, and this will involve (at least temporarily) giving up some of these more fanciful connections. In cooking terms: stop trying to make soufflΓ©s when you haven't yet mastered boiling water.
• $\binom{n}{k}$ does not "morph" into $\binom{n}{n-k}$. There is no transformation from one to the other. They are symmetric because the things they represent are symmetric. Brining topology or category theory into this is a complete red herring. As Hank Scorpio has suggested, you really need to learn the basics first, before you start going off on these flights of fancy.