From: Michael Barr <barr@triples.Math.McGill.CA>
In the category of chain complexes (over some abelian category) it is clear that if you identify homotopic arrows, you will also invert homotopy equivalences. Does anyone know if the converse is true? If you invert homotopy equivalences, do you wind up identifying homotopic arrows? What if you replace homotopy by homology? Does inverting maps that induce isomorphism on homology have the effect of identifying maps that induce the same map on homology?
Michael
In the category of chain algebras, this has been well studied the point being that the inverse (up to homotpy) of a multiplicative homotopy equivalence is generally NOT mutiplicative the new category is known as DASH and has an extensive literature jim