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Aug 23, 2019 at 0:26 comment added Tim Campion Indeed, by subdivision one can represent any homotopy type as the classifying space of a poset, which is both left and right cancellative. But far from being a monoid.
Aug 22, 2019 at 21:51 comment added Simon Henry The category of elements of a semi-simplicial sets is left cancelative. So I don't know for monoid, but at least one can represent any homotopy type as the realization of a left (or right) cancelative category by taking the category of elements (or its opposite) of any semi-simplicial sets representing it.
Aug 21, 2019 at 20:05 answer added Phil Tosteson timeline score: 5
Aug 21, 2019 at 19:36 history asked Tim Campion CC BY-SA 4.0