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I read in Gallauer's notes on infinity categories, in "Warning 1.9." without proof, that the geometric realization of a simplicial set does not determine it and wanted to make sure I understand why.

A proof of this statement should somehow construct a pair of non-isomorphic simplicial sets having isomorphic CW complexes. I claim however that if a simplicial set is a functor from the simplex category to sets, then both the data of the n-th simplices can be read from the CW complex, and the data of the various face and degeneration maps is encoded in the CW complex. Does this not prove we have a bijection between the objects of each of these categories? What am I missing?

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Here the geometric realization is presumably being considered just as a topological space, not as a CW complex, so the relevant equivalence is homeomorphism, not isomorphism of CW complexes. So, for instance, the simplicial set $\Delta^1$ has geometric realization homeomorphic to $[0,1]$, but so does a simplicial set with two nondegenerate $1$-simplices with a vertex from one of them equal to a vertex to the other (corresponding to decomposing $[0,1]$ into two edges).

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  • $\begingroup$ This explains it. Thanks! $\endgroup$
    – kindasorta
    Apr 21 at 21:09

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