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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.

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up vote 8 down vote accepted

Let $G$ be any group and consider the category with objects $A$, $B$, and $C$, where $\operatorname{Hom}(A,A)=G$ and the only other non-identity maps are a map $A\to B$ and a map $A\to C$. The nerve of this category has the homotopy type of $\Sigma BG$, which is simply connected and noncontractible as long as $G$ has nontrivial homology. But any quotient of this category that collapses all isomorphisms must be contractible.

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The closest thing to what you describe is Rezk's nerve $NC=N(C,\operatorname{iso} C)$. This is a complete segal space where in some sense isomorphisms are equivalent to identities.

See his paper.

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