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.


2 Answers 2


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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.