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This is a follow up to this question.

Imagine there is a finitely presented crossed module $\mathcal{G} = (G,H, -\triangleright-\colon G \to \operatorname{Aut}(H), \delta\colon H \to G)$ which I don't know. But for every other finite crossed module $\mathcal{G}'$, I know the number of homomorphisms $\mathcal{G} \to \mathcal{G}'$.

With these numbers, can I recover $\mathcal{G}$ up to isomorphism?

The motivation behind this is to understand the strength of the Yetter model. It counts homotopy equivalence classes from a 2-type to a fixed 2-type. Then a natural question is: "Given a manifold with an unknown homotopy 2-type, how much information about this 2-type can we recover from it through the Yetter model?"

By "finitely presented" or "finite" I mean that $G$ and $H$ are finitely presented, respectively, finite.

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  • $\begingroup$ If you don't know $\mathcal{G}$ then you cannot know what `another' crossed module might be, i.e., you cannot say: for every other finite crossed module. Have you tried thinking of this for groups rather than crossed modules? That might give some insight. $\endgroup$
    – Tim Porter
    Commented Jun 30, 2017 at 14:35
  • $\begingroup$ I do not see how knowing the number of homomorphisms helps. The Yetter model involves homotopy casases of maps not homomorphisms themselves. $\endgroup$
    – Tim Porter
    Commented Jun 30, 2017 at 14:38
  • $\begingroup$ @TimPorter, it seems like I'm mistaken, but aren't homotopy equivalence classes of maps between 2-types the same like homomorphisms between the corresponding 2-groups/crossed modules? The case for just groups is the question I linked at the top, i.e. this one. $\endgroup$
    – Manuel Bärenz
    Commented Jun 30, 2017 at 14:44
  • $\begingroup$ For the second part, I discovered that after I added the comment! For the first part, I do not know what you are saying. Homotopy classes of maps between two types are the same as homotopy classes of morphisms between the corresponding 2-groupoids. (You might find looking at the papers of Joao Faria Martins on Yetter invariants and things like that to be of use. BTW I think the answer to your question is No, but I may be wrong! $\endgroup$
    – Tim Porter
    Commented Jun 30, 2017 at 14:47
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    $\begingroup$ Also could you please define finitely presented crossed module? I think I know what you mean but others may not. $\endgroup$
    – Tim Porter
    Commented Jun 30, 2017 at 17:57

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I cannot be sure that there is no cunning method to do what you say but here is a comment about the problem.

The numbers you mention only give the dimensions of various vector spaces of the Yetter model, they do not tell you how these vector spaces are related by the functoriality of the TQFT concerned. It seems to me that this could be viewed from the perspective of Yoneda lemma type results. Suppose $\mathcal{C}$ is a category and $C$ an object, then you can get all the needed information on $C$ from the functor $\mathcal{C}(C,-): \mathcal{C}\to Sets$, but you cannot hope to do this by knowing just the sets $\mathcal{C}(C,D)$, you do need the functors concerned, or rather it would be interesting to characterise those categories $\mathcal{C}$ in which knowledge of those sets was sufficient.

You might argue that the fact of working with homotopy classes / natural transformations / conjugacy of maps would tell you something about the functors but you are not even encoding the groupoid of maps under homotopy, merely the $\pi_0$ of that, if you see what I mean, so that is not going to get you very far.

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    $\begingroup$ ...you cannot hope to do this just by knowing just the sets... Or worse, knowing just the cardinalities $|\mathcal{C}(C,D)|$ $\endgroup$
    – David Roberts ♦
    Commented Jul 1, 2017 at 8:35

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