Homotopy split monomorphisms

Question

Let $C$ be a model category. Recall that a morphisme $f : X \to Y$ is called a homotopy monomorphism if the diagonal $X \to X \times^h_Y X$ induces an isomorphism in the homotopy category. Suppose that one has morphisms $i : X \to Y$, $p : Y \to X$ such that $p i = 1_X$. Is $i$ then a homotopy monomorphism (and dually $p$ a homotopy epimorphism) ?

Answer 1

This is false even in $\mathbf{sSet}$. For instance, take any non-contractible connected simplicial set $X$ and any morphism $i : \Delta^0 \to X$. There is a unique morphism $p : X \to \Delta^0$ and, of course, $p \circ i = \mathrm{id}$. But $i : \Delta^0 \to X$ cannot be a homotopy monomorphism: this happens if and only if the loop space of $X$ is (weakly) contractible, and since $X$ is connected, that happens if and only if $X$ is (weakly) contractible.