internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
For $C$ a cartesian monoidal category (a category with finite products), an internal ring or a ring object in $C$ is an internalization to the category $C$ of the notion of a ring.
Under some reasonable assumptions on $C$ that allow one to construct a (symmetric) monoidal tensor product on the category of abelian group objects $Ab(C)$ internal to $C$, a ring object can also be defined as a monoid object internal to that monoidal category $Ab(C)$.
Sometimes one might take this last point of view a little further, especially in certain contexts of stable homotopy theory where a stable (∞,1)-category of spectra is already something like an (∞,1)-category-analogue of a category of abelian groups. With the understanding that a symmetric smash product of spectra plays a role analogous to tensor products of abelian groups, monoids with respect to the smash product are often referred to as “$xyz$-rings” of one sort or another (as mentioned at “ring operad”). Thus we have carry-over phrases from the early days of stable homotopy theory, such as “A-∞ rings” (for monoids) and “E-∞ rings” (commutative monoids). Here it is understood that the monoid multiplication on spectra is an $(\infty, 1)$-refinement of a multiplicative structure on a corresponding cohomology theory, with various forms of K-theory providing archetypal examples.
Let $T$ be the Lawvere theory for rings, viz. the category opposite to the category of finitely generated free rings (which are non-commutative polynomial rings $\mathbb{Z}\langle X_1, \ldots, X_n\rangle$) and ring maps between them. Then for $C$ a category with finite products, a ring object in $C$ may be identified with a product-preserving functor $T \to C$.
The more traditional definition, based on a traditional presentation of the equational theory? of rings, is that a ring object consists of an object $R$ in $C$ together with morphisms $a: R \times R \to R$ (addition), $m: R \times R \to R$ (multiplication), $0: 1 \to R$ (zero), $e: 1 \to R$ (multiplicative identity), $-: R \to R$ (additive inversion), subject to commutative diagrams in $C$ that express the usual ring axioms.
A ring object in Top is a topological ring.
A topos equipped with a ring object is called a ringed topos, see there for more details.
The affine line (see there) is a ring object in the given ambient topos.
ring, ring object
Last revised on March 22, 2021 at 04:23:40. See the history of this page for a list of all contributions to it.