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I tried to brush-up the entry fundamental infinity-groupoid which had been in a rather sorry state. Some things I did:
stated the definition (!) $Sing X$
removed Ronnie’s remark that there is a problem with this definition due to lack of chosen fillers and instead added Thomas Nikolaus’s theorem that when you choose fillers to get an algebraic Kan complex $\Pi X$ there is (still) a direct proof of the homotopy hypothesis
made the statement that $Sing X$ is equivalently computed by the abstract $\infty$-topos-theoretic definition of fundamental $\infty$-groupoid a formal proposition.
That looks good. I added a comment before the definition to the effect that we’re choosing Kan complexes as a notion of $\infty$-groupoid in order to make that definition.
polished up these two bibitems:
Keith A. Hardie, Klaus H. Kamps, Rudger Kieboom, A homotopy 2-groupoid of a Hausdorff space. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures 8 (2000) 209-234 [doi:10.1023/A:1008758412196]
Keith A. Hardie, Klaus H. Kamps, Rudger Kieboom, A Homotopy Bigroupoid of a Topological Space, Applied Categorical Structures 9 (2001) 311-327 [doi:10.1023/A:1011270417127]
Will give them an entry fundamental 2-groupoid now (which has been missing all along, even as a redirect)
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