symmetric monoidal (∞,1)-category of spectra
Ring is the category of rings (with unit) and ring homomorphisms (that preserve the unit).
A ring is a monoid in Ab, where $Ab$ is the category of abelian groups. So, $Ring$ is an example of a category of internal monoids.
Every surjective homomorphism of rings is an epimorphism in $Ring$, but not every epimorphism is surjective.
A counterexample is the defining inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ of the ring of integers into the ring of rational numbers. This is an injective epimorphism of rings.
For more see for instance at Stacks Project, 10.106 Epimorphisms of rings.
Last revised on September 9, 2020 at 07:56:47. See the history of this page for a list of all contributions to it.