okay back to your regularly scheduled polemic:
whoever the FUCK said that a topological space ("nice" or not whatever idc) presents a homotopy type, you were SADLY MISTAKEN
cuz there's actually two =)
one of them is (pre-)sheafy, and the other one is co-(pre-)sheafy =0
Conversation
ooh are you talking about the localic one you get from local systems?
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ummm idk am I? that sounds similar to what im doing but i dunno.
tldr if u dont wanna click through is: u look at all cech nerves of open covers of X. These fit into a cofiltered diagram and that should represent something like a pro-(∞-groupoid) (bit fuzzy on the details lmao)
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oh interesting, i saw the reply that said this had to do with shape theory but it’s not a construction i know anything about. also hmm maybe the “localic” is me confusing two things. what i had in mind is to pass from a space to local systems of oo-groupoids on that space
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and from there to apply a reconstruction theorem to get some kind of oo-groupoid out - maybe this is just shape stuff again
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hmm i don't understand what you're referring to enough to really be able to compare...
but fwiw all of this is basically motivated by
well
1) twitter.com/schala163/stat
but more generally like
2)
Quote Tweet
thinking about how to make homotopical sense of local triviality again...
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Replying to
it’s related - to talk about local systems you need to talk about sheaves being locally trivial. like in the 1-truncated case i’m saying take a space and consider the topos of locally trivial sheaves of sets on it; i think a reconstruction theorem spits a groupoid back out...
...which is a version of the fundamental groupoid and which probably has something to do with the cech homotopy type but who knows, i’ve never tried to calculate this thing
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hmmm okay I think im following
I would bet that that is pretty much identical to calculating the cech homotopy type when you unpack everything...
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