Let $X$ be a based connected CW-complex $X$.
Say that $X$ is CW-flat if, for each map of based connected CW-complexes $f \colon Y \to X$ such that $\pi_n(f)$ is injective, $\pi_n X \wedge f$ is also well.
How does CW-flat compare to flatness? Are there any examples? Are infinite loop spaces examples of CW-flat CW-complexes?
CW-flatness is interesting for another thing because of the internal groupoid structure associated to $([X \wedge X,Y],[X \wedge X \wedge X,Y])$ for each based connected CW-complex $Y$. This is something which in the situation for rings, e.g. $([K \otimes_{\mathbb{Q}} K,K], [K \otimes_{\mathbb{Q}} K \otimes_{\mathbb{Q}} K,K])$ is essentially $\mathrm{Gal}_{\mathbb{Q}}(K)$ for a Galois field extension $K$ of $\mathbb{Q}$.
Smash is a pre-cursor to smash product of ∞-spaces and of spectra, but it is harder to find sources for the nature of the algebra involved (e.g. internal monoids in the case of locally compact Hausdorff spaces).
In homological algebra one proves that there are enough projectives and injectives, and projectives are flat, so that there are enough flats to perform replacement and other calculations. So my main question is about the analogue of Tor for the case of smash product of based spaces.