The question is motivated by Eckmann-Hilton duality and certain flaws of the homotopy category of CW-complexes. Unfortunately, I do not know the formalism of model categories, so excuse me if it is a basic fact concerning them.

First consider a circle $S^1$ with a fixed point. All spaces are supposed to be connected and have a fixed point, and I will suppress it in notations. Then there are its $(n-1)$-suspension $S^n:=\Sigma^{n-1} S^1$ and its $(n-1)$-delooping $K(\mathbb Z, n):=\Omega^{-(n-1)}S^1$ (I believe that one can construct it geometrically). From them we can build $n$-th homotopy groups $\pi_n(X):=[S^n, X]$ and $n$-th (integral) cohomology groups $H^n(X):=[X, K(\mathbb Z, n)]$, where $[A, B]$ means a set of homotopy classes of maps from $A$ to $B$, and give them a natural group structure, see Fomenko, Fuchs Homotopical topology, $\S$1.4.

As suspension $\Sigma$ and looping $\Omega$ are adjoint, one has $H^i(S^n) \simeq \mathbb Z$ for $i=n$ and $0$ otherwise, and $\pi_i(K(\mathbb Z, n)) \simeq \mathbb Z$ for $i=n$ and $0$ otherwise (we ignore $\pi_0$ and $H^0$). Moreover, in the category of CW-complexes $K(\mathbb Z, n)$ is defined by this condition up to homotopical equivalence, but $S^n$ is not -- there is Poincaré homology sphere $\widehat{S^3}$.

A quick idea is that it happens because CW-complexes are constructed from $S^n$, so one should work in the category of spaces constructed from $K(\mathbb Z, n)$, with something like Postnikov towers instead of CW-complexes' skeletons. So a question is whether such a thing exists and has nice properties?