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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 $\kappa$.

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  • $\begingroup$ Naive question: what the status of the class of finite CW complexes regarding question 1 ? Is it known they are not enough, or you don't know ? $\endgroup$
    – Simon Henry
    Commented Jan 31, 2020 at 19:55
  • $\begingroup$ @SimonHenry I don't know -- but I imagine that somebody does! If finite CW complexes don't work, it's hard to imagine that anything will! Maybe I should add that as an additional question. $\endgroup$
    – Tim Campion
    Commented Jan 31, 2020 at 19:56
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    $\begingroup$ By the way, Rosicky’s paper had an error invalidating the main claims, which remain open as far as I know. The Arxiv version has been updated: arxiv.org/pdf/math/0506168.pdf $\endgroup$
    – Kevin Carlson
    Commented Feb 1, 2020 at 1:02
  • $\begingroup$ @KevinCarlson Ah, thanks! I was about to use that result for something! $\endgroup$
    – Tim Campion
    Commented Feb 1, 2020 at 1:12
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    $\begingroup$ A counter-example to my "theorem" is in arxiv.org/pdf/1102.3240.pdf. $\endgroup$
    – Jiří Rosický
    Commented Feb 1, 2020 at 16:40

1 Answer 1

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This paper by Kevin Carlson and Dan Christensen says that the answer to question one is no: No set of spaces detects isomorphisms in the homotopy category, arXiv:1910.04141.

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  • $\begingroup$ Ah, perfect! (Though rather disappointing.) Thanks! $\endgroup$
    – Tim Campion
    Commented Jan 31, 2020 at 20:23
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    $\begingroup$ @TimCampion it's worth pointing out that in another paper Kevin Carlson shows that there is a 2 categorical generating set. Maybe he'll chime in to say more. arxiv.org/pdf/1802.04439.pdf $\endgroup$
    – J Cameron
    Commented Jan 31, 2020 at 20:26
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    $\begingroup$ @TimCampion It all comes down to the existence of noninvertible group homomorphisms which, according to groups of bounded size, appear to be conjugate to the identity. In the 2-morphisms of the homotopy 2-category we get our hands on homotopies with endpoints fixed, repairing the mod-conjugacy problem and allowing the spheres to weakly generate. The homotopy 2-category of any presentable infinity category also gets a weak generator. I find it mysterious, spaces not being an $(\infty,2)$-category, that their homotopy 2-category should be the nice thing, but so it goes. $\endgroup$
    – Kevin Carlson
    Commented Feb 1, 2020 at 0:55
  • $\begingroup$ Carlson's paper is called On Whitehead’s theorem beyond pointed connected spaces (arXiv:1802.04439). @Kevin that result is remarkable to me. Why tori? Is there a good conceptual reason? $\endgroup$
    – David Roberts ♦
    Commented Feb 1, 2020 at 8:56
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    $\begingroup$ @DavidRoberts There’s nothing very special about the tori here. It’s just that the family of maps $\pi_1 L^n f$, $L$ the free loop space functor, see how $f$ acts on $\pi_n$ at each base point, and can be seen in the homotopy 2-category. I find it less intuitive to work with unbased mapping spaces from spheres, but Raptis convinced me that in fact the spheres are a generator as well. $\endgroup$
    – Kevin Carlson
    Commented Feb 1, 2020 at 12:13

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