Do spaces admit a weak cogenerating set? Let $\mathcal C$ be a category. Say that a class of objects $\mathcal S \subseteq \mathcal C$ is weakly cogenerating 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 bijections $Hom_C(Y,S) \to Hom_C(X,S)$ for every $S \in \mathcal S$.
Of course, every category $C$ admits a weakly cogenerating class -- namely, take $\mathcal S = \mathcal C$. But it's frequently important to have a cogenerating set -- i.e. to require that $\mathcal S$ is small.
Question: Does the homotopy category (of spaces) admit a weak cogenerating set?
It's clear that the homotopy category of simply-connected spaces admits a weak cogenerating set -- we can take $\mathcal S = \{K(\mathbb Z, n) \mid n \geq 2\}$ or alternatively $\mathcal S = \{K(k,n) \mid n \geq 2, k \in \{\mathbb Q, \mathbb F_p\}\}$ in this case by the cohomology Whitehead theorem. But I'm pessimistic about the chances of doing something similar with arbitrary spaces.


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*Relatedly, I wonder whether the category of groups admits a weak cogenerating set.

*I also wonder whether the class of truncated spaces -- those spaces $S$ for which $\pi_k(S) = 0$ for $k$ sufficiently large -- is a cogenerating class for the homotopy category. What about the class of Eilenberg-MacLane spaces?
 A: For any infinite set $X$ let $S_X$ be the group of bijections $\sigma \colon X\to X$ such that $\{x : \sigma(x)\neq x\}$ is finite.  This still has signature homomorphism, and the alternating subgroup $A_X$ is simple, and has the same cardinality as $X$.  Now let $\mathcal{G}$ be a set of groups, and put $\kappa = \max \{|G|:G\in\mathcal{G}\}$.  Then $\text{Hom}(A_X,G)$ will be a singleton for all $G\in\mathcal{G}$ and $X$ with $|X|>\kappa$ (because the kernel of any homomorphism is nontrivial by cardinality, and so is the whole of $A_X$ by simplicity).  So $\mathcal{G}$ is not a weak cogenerating set.  
It doesn't seem to be straightforward to deduce the corresponding result for the homotopy category.
EDIT To summarize the discussion in the comments, we can indeed deduce the corresponding result for the homotopy category with a little more work. Choose an acyclic simple group $G$ bigger than the fundamental group of any space in $\mathcal S$. Then any map $f: BG \to S$ for $S \in \mathcal S$ is trivial on $\pi_1$ by simplicity, so it lifts to the universal cover $\tau_{\geq 2} S$. By acyclicity, the composite map $BG \to \tau_{\geq 2} S \to K(\pi_2(S),2)$ is trivial so $f$ lifts through the 2-connected cover $\tau_{\geq 3} S$. Continue in this manner, lifting through the Whitehead tower to see that $f$ is nullhomotopic. Thus $\mathcal S$ does not distinguish $BG$ from a point, and is not weakly cogenerating.
