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.


* 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?



Do spaces admit a weak cogenerating set?
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](https://mathoverflow.net/questions/348640/do-there-exist-acyclic-simple-groups-of-arbitrarily-large-cardinality/348643#348643) 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.


