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For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.
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How many simplicial complexes on n vertices up to homotopy equivalence?
Andrew Newman just posted a preprint to the arXiv showing that the answer is doubly exponential in $n$.
In particular, he showed that the number of homotopy types of simplicial complexes on $n$ verti …