@AdrianClough Thank you for the questions. I just want to mansion that, for sheaves of p-complete spectra, many things about sheaves versus hypersheaves where done here: arxiv.org/abs/1905.06611.
Now, I'm probably unable to answer any questions, but your answer seemed very strong -- there are definitely people working on persistent homology that are in homotopy theory research groups
@HarryGindi I would suggest that as a general rule we should be as open-minded as possible about what gets discussed in the chat here. If somebody asks a question of the form "Is it okay to discuss mathematical topic X here?", the answer should (almost) never be "no". At worst, the answer should be "It seems unlikely that folks will have much to say about X, but give it a shot!".
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The only marginally-plausible scenario I can think of is if somebody repeatedly asks about X with nobody taking them up on the topic of conversation. In such a case, a gentle suggestion that people don't seem particularly interested in X here might be in order.
I'd like to avoid too much "gatekeeping".
In this spirit, @joe345wa I'd say this chat is not chock-full of experts on persistent homology, but you're certainly welcome to bring it up here!
Of course, when I say we should be open-minded, I mean this in the context of mathematical discussions. I would agree that we should be a bit more choosy about what non-mathematical things we discuss.
Thanks @TimCampion my question is perhaps actually more related on the tools available rather than the PH itself, so it might still be OT. Essentially, I can see that many efficient tools for computing persistent homology and persistent diagrams exist. They usually works on point clouds and/or images. However I couldn't find a good way to get the actual clusters of a certain homology class given a filtration threshold.
Basically, given an HC, I need the clusters (i.e. sets of points) that are alive (i.e. not merged) at a certain time $t$. To my understanding this should be one of the most common use cases, and yet I couldn't find an easy way to get said clusters from the tools I've checked. I am not an expert of TDA, so I suspect that there's something in the PH that I'm completely missing. That's because to my understanding those clusters must be computed in order to create the persistence diagram.