Let $C$ be a (small) category. One can form the free groupoid $GC$ of $C$ which is the left adjoint construction to the inclusion functor $\mathrm{Groupoid}\rightarrow\mathrm{Category}$. Is then $C$ always homotopy equivalent to $GC$? In other words, are the spaces $BC$ and $B\pi BC$ homotopy equivalent where $B\underline{}$ is the classifying space and $\pi\underline{}$ is the fundamental groupoid?



Free groupoid and homotopy equivalence
No. There is a monoid with trivial group image whose classifying space is a sphere. See [Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?](https://mathoverflow.net/questions/122221/is-there-a-discrete-monoid-m-injecting-into-its-group-completion-g-for-which-bm)


Basically take the idempotent semigroup with elements $(a,b)$ with a,b either 0 or 1 and multiplication is (a,b)(c,d)=(a,d). Next add an identity. Clearly the fundamental group of this idempotent monoid is trivial. It is known to have classifying space homotopic to a 2-sphere. 


