Are weak homotopy equivalences always homotopic? I was wondering whether two weak equivalences between spaces $X$ and $Y$, are homotopic, or not. If yes, a proof / reference would be very welcome. If not, a counterexample would be interesting!
More generally, I am interested in the case of any category with a notion of weak equivalences / homotopies between weak equivalences, but I think the archetypal case of topological spaces is a good way to start.
Have tried a few things so far, but nothing worth mentioning yet.
 A: No, the identity map $S^2 \to S^2$ and the antipodal map are self-homotopy equivalences of $S^2$, yet the two maps are not homotopy equivalent.
A: The situation is of course far more ridiculuous than you could possibly hope for.
For instance take the paper Every finite group is the group of self-homotopy equivalences of an elliptic space, Acta Math. 213, 1, (2014), 49-62, by C. Costoya, A. Viruel. If you have not yet guessed its content:

Every finite group can be realised as the group of self-homotopy equivalences of a simply-connected CW complex.

In fact the authors show that for any given finite group there are infinitely many such distinct complexes with the stated property, and moreover that each can be taken as the rationalisation of a simply-connected manifold. Even more details are included in the paper, and anyone interested in rational homotopy theory should enjoy skimming through it.
To include some details for the beginner, the 'group of self-homotopy equivalences' of $X$ is the set of all homotopy classes of homotopy self-equivalences $X\xrightarrow\simeq X$ given the composition product. Distinct elements of this group are distinct homotopy classes of maps. If $X$ is (homotopy equivalent to) a CW complex then there is no distinction between homotopy self-equivalence and weak homotopy self-equivalences.
A: For a very simple counterexample, take $X$ and $Y$ to be discrete spaces of the same cardinality.  Then any bijection between them is a homotopy equivalence (indeed, a homeomorphism), but there are no nontrivial homotopies of maps $X\to Y$.
A: A counterexample is given by considering the antipodal map and the identity map $S^1\to S^1$. Both are weak equivalences, but certainly not homotopic.
