In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"?
The Unicode standard lists all of them inside the Mathematical Operators Block.
On
≈ is for numerical data, homeomorphism
≃ is for homotopy equivalence
≅ is for isomorphism, congruence, etc
These are just my own conventions.
On
As long as the category theory is concerned:
The symbol ≅ is used for isomorphism of objects of a category, and in particular for isomorphism of categories (which are objects of CAT). The symbol ≃ is used for equivalence of categories. At least, this is the convention used in this book and by most category theorists, although it is far from universal in mathematics at large.
(Warning 1.3.16, Basic Category Theory by Tom Leinster).
On
In my work "=" is the identity of a number so it states an equivalence. 1=1, 2x=10 ie x =5.
The approximation sign "≈" I use for decimal approximations with tilde "~" being a rougher approximation.
Tilde "~" I use to state a geometric shape is similar to another one ie a triangle of sides 3/4/5 is similar to a triangle with sides 30/40/50. I write ▲ABC ~ ▲A'B'C' where ▲A'B'C' is a dilated version of the pre-image.
For a closer similarity "≃" might mean a triangle almost congruent but only ROUGHLY similar, such as two triangles 3/4/5 and 3.1/4.1/5.1 while "≅" means congruent. Real life triangles use approximations and have rounding errors. 3/4 does not equal 3.1/4.1 but could be rough approximations for something already constructed. As such this is not a popular use and purists and rigorous math profs disdain it because they do not have a way of using it or defining it soundly. That is why laymen or service professionals are free to explore it and academics prefer something more clearly defined unless deformations are allowed as in my case.
Triangle 3/4/5 is ≅ to triangle 4/5/3 the difference being a rotation changing the coordinates of the angles but preserving angle and side length.
Also I use the definition symbol "≡" to define functions. Simplest case is f(x) ≡ y. I am using it to state a relationship to find sums instead of stating it as the sum that we are deriving however it can be used to state equivalence. For a quadratic I would set f(x) to zero but would not define f(x) as zero. I use ≡ for all cases, not only immediate ones. Setting f(x) to zero creates the equivalency f(x) = 0 for the coordinate you are trying to solve but is not true for all coordinates that are solvable.
The notations $\cong$ and $\simeq$ are not totally standardized. Both are usually used for "isomorphic" which means "the same in whatever context we are." For example "geometrically isomorphic" usually means "congruent," "topologically isomorphic" means "homeomorphic," et cetera: it means they're somehow the "same" for the structure you're considering, in some senses they are "equivalent," though not always "equal:" you could have two congruent triangles at different places in a plane, so they wouldn't literally be "the same" but their intrinsic properties are the same. I've seen colleagues use both for isomorphic, and some (mostly the stable homotopy theorists I hang out with) will use $\cong$ for "homeomorphic" and $\simeq$ for "up to homotopy equivalence," but then others will use the same two symbols, for the same purposes, but reversing which gets which symbol.
The $\approx$ is used mostly in terms of numerical approximations, meaning that the values in questions are "close" to each other in whatever context one is working, and often it is less precise exactly how "close." Topologists also have a tendency to use $\approx$ for homeomorphic.
The main take-away from this answer: notation is not always standardized, and it's important to make sure you understand in whatever context you're working.