> you are coming across as someone who is trivializing a potentially important piece of work.

I doubt it. The parent is confused and hence the question obviously. But I get the impression that the parent would benefit from reading an elementary textbook on discrete math as the post is gibberish (as are a few other posts in this thread). I know it sounds rude, but I don't know how else to put it. It's not only the fact that the parent doesn't have a conceptual idea about equivalence relations and in particular equality, isomorphism, let alone homotopy, but is also very confused about what it means for a series to converge which is also very elementary. Before discussing the merits of Dostoevsky in Russian, one must learn and know the Russian alphabet at the very least or something.

What you have up there is an equality. For example, x = y means x and y are the exact same thing. Equality is one example of equivalence relation of which there are a ton. Equivalence relation is a more general concept.

A set is a bag of items. We usually call these items elements of a set. For example, {1, 2, 4} is a set containing elements 1, 2, 4.

Ordered pairs are tuples written like so: (a, b) where (a, b) != (b, a).

Relation is an association from one set to another. For example, a -> x, b -> &, c -> apple is a relation from {a, b, c} to {x, &, apple}. Let R stand for this relation, then we write R = {(a, x), (b, &), (c, apple)} where R associates a with x, b with & and c with apple.

Consider the sets X = {1, 2, 4, 5} and Y = {6, 9, 1000, 5} with the following association R from X to Y: 1 -> 9, 2 -> 6, 4 -> 1000, 5 -> 5. This R looks like an ordering relation meaning this R is ≤. Let's see, 1 ≤ 9, 2 ≤ 6, 4 ≤ 1000, 5 ≤ 5. Then ≤ = {(1, 9), (2, 6), (4, 1000), (5, 5)}.

Some relations are called functions if they meet certain criteria.

If a relation R abides by the following rules it is called an equivalence relation. For any a, b, c in a set S:

1. (a, a) in R

2. If (a, b) in R, then (b, a) in R

3. If (a, b) and (b, c) in R, then (a, c) in R.

For example, congruence modulo integer n is an equivalence relation as it abides by the three rules above which can be easily shown. Two integers a, b are congruent modulo n if n divides a - b.

Equality is but one of many different equivalence relations.

What I wrote above doesn't even come close to scratching the surface of this topic. However, any introductory textbook on abstract algebra, discrete math or elementary number theory will have this material well fleshed out. Also, every intro to proofs book dedicates significant number of pages to the topic of relations. But this stuff is very elementary and many steps removed from category theory and so you are missing some foundational background to understand the topic broached in the article. You want to have some experience with abstract algebra and general topology before approaching category theory.

As for the equality you gave above,

let X = 1 + x + x^2 + ... and Y = 1/(1 - x). Then X is a geometric series and X = Y only if |x| < 1. If |x| >= 1, then X diverges. When X = Y, you cannot talk about one side of the equality diverging and the other not as they both are the same object.

This topic of geometric series (and many others) can be found in any textbook on elementary real analysis. Textbooks are great because they start from the beginning, don't leave out (or add) some random topics, give airtight definitions and the statements are given watertight proofs (like that one above about X = Y needs proof).