When this subject comes up I always think that the equality sign is loaded. It can mean, definition, equality, identity, and proportionality. How come this state of the equality sign is not mentioned in the paper?
(x + y)(x - y) = x^2 - y^2
(an identity, since it’s true for every x and every y) and something more arbitrary like
3x^2 + 2x - 7 = 0
(an equation certainly not valid for all x and whose solutions are sought).
Of course, really, the first one is a straightforward equality missing some universal quantification at the front… so maybe that’s just what the triple equals sign would be short for in this case.
and I suppose the model of localization discussed in the article above counts as well.
With my font, the letter l looked like a vertical bar |. Hence why this looked like symbol soup. But okay, let's say it's about projective spaces. Then surely, you realize that the equal sign still denotes equality. The point [1:2] is literally equal to [2:4] in P^2, not "proportional". Perhaps you need a refresher: https://en.wikipedia.org/wiki/Equivalence_class#quotient_set
I later remembered fractions a/b = ac/bc work this way too!
:= definition ≡ identity = equality ∝ proportionality
= definition = identity = equality = proportionality