It is well known that a simply connected groupoid is already contractible. Thus, isomorphisms cannot model higher homotopy. But I wonder, is this a global phenomenon (because we consider categories with isomorphisms only) or is it only local. What I mean with the latter is the following.

Instead of a simply connected groupoid, consider a simply connected category $\mathcal{C}$. Can we somehow contract the isomorphisms in $\mathcal{C}$ to points without changing the homotopy type of $\mathcal{C}$?

A bit more precisely: Consider two objects in $\mathcal{C}$ as equivalent if there is an isomorphism betweem them. Is it possible to define furthermore an equivalence relation on the arrows such that we obtain a quotient category homotopy equivalent to $\mathcal{C}$ and isomorphisms in $\mathcal{C}$ are identified with points (identities).

At least this is possible if we assume that the subcategory consisting of all the isomorphisms in $\mathcal{C}$ is a contractible groupoid. But this is not the general case.