In ordinary category theory, coequalisers are always epimorphisms, so effective epimorphisms are indeed epimorphisms. But this is not so in higher category theory.

Indeed, consider the unique morphism from the circle *S*^{1} to the point Δ^{0}. This is surjective on connected components, so it is an effective epimorphism in the (∞, 1)-category of spaces. But it is not an epimorphism: indeed, a morphism *X* → Δ^{0} is an epimorphism in the (∞, 1)-category of spaces if and only if the (unreduced) suspension of *X* is (weakly) contractible, and the suspension of *S*^{1} is the sphere *S*^{2}, which is certainly not contractible.

On the other hand, there are non-trivial epimorphisms *X* → Δ^{0} is the (∞, 1)-category of spaces, i.e. there is a space *X* whose (unreduced) suspension is (weakly) contractible: see here for details.

[Thanks to Aaron Mazel-Gee and Saul Glasman for helping me figure this out.]

### Like this:

Like Loading...

*Related*