Group objects detected on path components

Proof. If \(M\) is a group, then applying \(\pi _0\) to its shear isomorphism shows that the shear map of \(\pi _0(M)\) is a bijection, so \(\pi _0(M)\) is a group. Conversely, suppose that \(\pi _0(M)\) is a group. For every point \(x\in M\), choose a point \(y\in M\) whose component is inverse to that of \(x\). The products \(xy\) and \(yx\) lie in the unit component, so multiplication by \(y\) is a homotopy inverse to multiplication by \(x\). The shear map \[ (\pr _1,m)\colon M\times M\longrightarrow M\times M \] is a map over the first factor whose fiber at \(x\) is multiplication by \(x\). It is therefore an isomorphism, and hence \(M\) is a group. □