A higher dimensional homotopy sequence (Q1605627)

Given a continuous map \(f : X \to Y\) of pointed spaces, there is a well known long exact sequence relating the homotopy groups of \(X\), \(Y\) and \(Kf\), the homotopy kernel of \(f\). This note shows that this is in fact the projection of a `2-exact' (in a suitable sense) sequence of some homotopy pointed groupoids. The paper not only develops the machinery for this 2-exact sequence, but explores various aspects of a strong form of this notion of 2-exactness.