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
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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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