I am working on Weibel's K-book and have, while meditating phantom maps and weak homotopies, asked myself if in the same way as their is a category $Ho(Top)$ identifying homotopic maps, is their a category $Ho_w(Top)$ identifying weakly homotopic maps?
To recall, two maps are weakly homotopic if they induces homotopic maps when precomposed by any map from a finite CW-complex. I do not know any of the formalism of model category theory (which I believe is the setting for precisely defining $Ho(Top)$) which implies my question might be "bad" in the sense that the answer is simply "yes you can, just do it".
Thank you for any help in satisfying my curiosity.
The model theory people probably have a smarter answer, but to "naively" answer the question: Yes.
In general, if $\mathcal{C}$ is a category and we have an equivalence relation $\sim$ on the Hom-set $\mathcal{C}(x,y)$ for all objects $x,y\in\mathrm{Ob}(\mathcal{C}))$ s.t. $f\sim f^{\prime}$ in $\mathcal{C}(x,y)$ and $g\sim g^{\prime}$ in $\mathcal{C}(y,z)$ implies $gf\sim g^{\prime}f^{\prime}$ in $\mathcal{C}(x,z)$, then we can form a quotient category $\mathcal{C}/\sim$ with the same objects as $\mathcal{C}$ and Hom-sets $(\mathcal{C}/\sim)(x,y)=\mathcal{C}(x,y)/\sim$. The assumption on $\sim$ implies that composition in $\mathcal{C}$ induces a well-defined composition on $\mathcal{C}/\sim$ and the lattter inherits the axioms of a category from the former. Then, there is a projection functor $p\colon\mathcal{C}\rightarrow\mathcal{C}/\sim$, which is universal with the property that $f\sim f^{\prime}$ implies $p(f)=p(f^{\prime})$.
Taking $\mathcal{C}=\mathbf{Top}$ and $\sim$ the relation of being homotopic, we obtain the usual naive homotopy category whose objects are topological spaces and whose morphisms are homotopy classes of maps. Taking instead $\sim$ to be the relation of being weakly homotopic, we obtain your naive weak homotopy category whose objects are topological spaces and whose morphisms are weak homotopy classes of maps. It is only necessary to check that the relation of being weakly homotopic is an equivalence relation compatible with composition, but this is not difficult.