Does $\pi_1(S^1)\approx \mathbb{Z}$ mean the same thing as $\pi_1(S^1)=\mathbb{Z}$?

Question

In Hatcher's Algebraic Topology, one of the first results encountered is that the fundamental group of the circle, $\pi_1(S^1)$, is isomorphic to the integers $\mathbb{Z}$. notated in the book as $\pi_1(S^1)\approx\mathbb{Z}$.

However, in other documents I have found online (take the solution to problem 16 (a) in this document, for instance), I keep seeing the equality ($=$) symbol in the place of the isomorphic ($\approx$) symbol, i.e. instead of $\pi_1(S^1)\approx\mathbb{Z}$ it is written as $\pi_1(S^1)=\mathbb{Z}$.

So my question is: which is it? That is, what notation is technically correct? Or are they both correct?

Answer 1

There is a good article about 'equality': Mazur - When is one thing equal to some other thing.

Answer 2

Definitely not equality, just isomorphisim. Equality is a much stronger property (of sets) while isomorphism just means equivalence in that current category you're in (so groups for your case).