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.
For more see at Stacks Project, 10.106 Epimorphisms of rings.
Every surjective homomorphism of rings is an epimorphism in $Ring$, but not every epimorphism is surjective.
A counterexample:
In unital Rings, the canonical inclusion $\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}$ of the integers into the rational numbers is an epimorphism.
Since every rational number is the product of an integer with the multiplicative inverse of an integer
and since unital ring homomorphism
preserve multiplicative inverses, $f\left( a/b \right) = f(a) \cdot \big(f(b)\big)^{-1}$, it follows that any pair $(f,g)$ of parallel morphisms on $\mathbb{Q}$ are equal as soon as they take equal value on the integers.
Last revised on January 21, 2021 at 23:59:20. See the history of this page for a list of all contributions to it.