As a follow-up to this question, let $\mathcal C$ be a category and $\mathcal S \subseteq \mathcal C$ a class of objects. Say that $\mathcal S$ is *weakly generating* if the functors $Hom_{\mathcal C}(S,-)$ are jointly conservative, for $S \in \mathcal S$. That is, a map $X \to Y$ in $\mathcal C$ is an isomorphism if and only if it induces a bijection $Hom_{\mathcal C}(S,X) \to Hom_{\mathcal C}(S,Y)$ for each $S \in \mathcal S$.

**Question 1:** Does the homotopy category of spaces admit a small generating set? (For example, as Simon Henry asks, do finite CW complexes work? How about the spheres?)

Of course, by Whitehead's theorem, the homotopy category of *pointed connected* spaces admits a generating set given by the spheres. But I'm not sure about unpointed spaces.

Note that the singleton set comprising the contractible space $\ast$ is a generator in the $\infty$-category of spaces, since $X \to Y$ is an equivalence if and only if $Map(\ast, X) \to Map(\ast,Y)$ is an equivalence. But passage to the the homotopy category discards the higher homotopy of the mapping spaces.

**Question 2:** More generally, if $\mathcal C$ is an accessible $\infty$-category, then does the homotopy category $h\mathcal C$ admit a small generating set? What if we assume that $\mathcal C$ is presentable?

Again, if $\mathcal C$ is $\kappa$-accessible, then the class $\mathcal C_\kappa$ of $\kappa$-compact objects forms a generating set in $\mathcal C$, but it's not clear if it forms a generating set in $h\mathcal C$. In fact, I think that Question 2 (in the "presentable" case) is equivalent to Question 1: if the answer to Question 1 is affirmative, so that $\mathcal S$ is a generating set for the homotopy category of spaces and $\mathcal T$ is a generating set for $\mathcal C$, then the set of spaces $S \ast T$ for $S \in \mathcal S, T \in \mathcal T$ forms a generating set for $h\mathcal C$. Here $\ast$ denotes copowering.

One result in this direction is Rosicky's Theorem, which says (in model-independent language) that if $\mathcal C$ is a presentable $\infty$-category, then the canonical functor $h\mathcal C \to Ind_\kappa(h\mathcal C_\kappa)$ is essentially surjective ~~ and full ~~ for some $\kappa$. For my purposes, it would suffice to know that this functor is conservative for some $\kappa$.