On the homotopy relation and self equivalences (Q1840587)

Let \(C\) be a category with a homotopy relation \(\simeq\) on its morphism sets \(C(X,Y)\) for each object \(X\) and \(Y\), which is compatible with composition in the sense that for \(f,g\in C(A,B)\) and \(c\in C(B,X)\), \(a\in C(Y,A)\), \(f\simeq g\) always implies that \(a\circ f\simeq a\circ g\) and \(f\circ b\simeq g\circ b\). In this paper, the author studies a refinement of the homotopy relation which is generated by composing maps with homotopy trivial self-equivalences of the objects, where \(f\in C(X,X)\) is called a homotopy trivial self equivalence if \(f\simeq \text{1}_X\). In general, this new equivalence relation on maps does not coincide with homotopy. The author shows that it coincides for the category of chain complexes over a field, for the category of maps of a simplicial sphere to a Kan simplicial set, and for the category of smooth maps from a closed smooth manifold to a sphere.