What is the [center][1] of the [homotopy category][2] $\mathbf{hTop}$? I strongly believe that it is trivial, but it is hard to prove since $\mathbf{hTop}$ is [not concretizable][3] and hence has no small [separator][4]. This also means that currently I have no proof why the center is even a set ...

The center of $\mathbf{hTop}$ consists of homotopy classes of continuous maps $\alpha_X : X \to X$ for every space $X$. For every continuous map $f : X \to Y$ there should be a homotopy between $\alpha_Y \circ f$ and $f \circ \alpha_X$. These homotopies are not subject to any compatibility conditions when $f$ changes.

Clearly, $\alpha_X$ is (homotopic to) the identity when $X$ is contractible. Also, $\alpha_{\coprod_{i \in I} X_i}$ identifies with $\coprod_{i \in I} \alpha_{X_i}$, so the same holds when $X$ is a coproduct of contractible spaces. The easiest space which is not of this form is the circle $S^1$. I have no idea how to approach $\alpha_{S^1}$.

I am open for $1$-categorical variations of the spaces, such as CW complexes, CGWH spaces or pointed spaces. But in this question I am not asking about higher categorical versions of the center.


  [1]: https://ncatlab.org/nlab/show/center
  [2]: https://ncatlab.org/nlab/show/Ho%28Top%29
  [3]: http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html
  [4]: https://ncatlab.org/nlab/show/separator