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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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...
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...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
Replying to
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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