We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \mathsf{Grp} \quad\text{and}\quad H_n : \mathsf{Top} \to \mathsf{Ab}$$ (actually this works even if the spaces are homotopy equivalent). The important thing here is that these functors can be used to prove that the two spaces are not homeomorphic: for instance $H_3(S^3) \cong \Bbb Z \not\cong 0 = H_3(S^2)$, so that $S^3$ and $S^2$ are not homeomorphic (they don't even have the same homotopy type).
I was wondering whether there was somehow a "converse" to this, i.e. is they a way to prove that two topological spaces are homeomorphic. More precisely:
Is there a category $\scr C$ and a functor $\mathsf F : \mathsf{Top} \to \mathscr C$ such that $\mathsf F(X) \cong \mathsf F(Y) \implies X \cong Y$ ? Of course, I want to avoid obvious examples as $\mathsf{Id_{Top}}$ .
(By the way, I don't know if there is a name for such functors, which are injective on objects. Faithful is already used for something different). I would also accept discussing the case where the homeomorphism $ X \cong Y$ is replaced by a homotopy equivalence $X \simeq Y$.
Trying the functor $\mathsf F(X) = X \times X$ doesn't work, as shown here. The functor $\mathsf F(X) = X \sqcup X$doesn't work as well.
The closest result I found is a theorem due to Gelfand and Kolmogorov : given two compact and Hausdorff spaces, if the commutative rings $C(X)$ and $C(Y)$ of continuous functions $f\,:\,X,Y\rightarrow \mathbb{R}$ (under pointwise addition and multiplication) are isomorphic, then $X$ and $Y$ are homeomorphic. Maybe we could try to generalize this to the category of locally compact and Hausdorff spaces, using the Alexandorff compactification.
Thank you for your comments!