# Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?

McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents.

Question: Which homotopy types are realized as the classifying space of a right-cancellative discrete monoid?

In the commutative case, my guess would be that $$BM \simeq B(M[M^{-1}])$$, so that the classifying space is aspherical. But I'm less confident that this happens in the noncommutative case.

• 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. – Simon Henry Aug 22 '19 at 21:51
• 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. – Tim Campion Aug 23 '19 at 0:26

By universal properties, we have that $$BM$$ is the classifying space of the homotopy localization $$B(M[M^{-1}]^h)$$. Thus $$BM$$ is aspherical if and only if the homotopy localization is discrete.
Further, Dwyer-Kan showed that if $$(M,W)$$ admits a calculus of fractions, then the homotopy localization agrees with the ordinary localization. When $$M$$ is cancellative, $$(M,M)$$ admits a calculus of fractions if and only if it satisfies the Ore condition: $$\forall m_1, m_2 \in M, \exists n_1, n_2 \in M, ~ n_1 m_1 = n_2 m_2.$$ So in this case $$M[M^{-1}]^h \simeq M[M^{-1}]$$ and the homotopy localization is aspherical. In general, I do not know what happens.