ring object



Category theory



For CC a cartesian monoidal category (a category with finite products), an internal ring or a ring object in CC is an internalization to the category CC of the notion of a ring.

Under some reasonable assumptions on CC that allow one to construct a (symmetric) monoidal tensor product on the category of abelian group objects Ab(C)Ab(C) internal to CC, a ring object can also be defined as a monoid object internal to that monoidal category Ab(C)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 “__-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 (,1)__(\infty, 1)-refinement of a multiplicative structure on a corresponding cohomology theory, with various forms of K-theory providing archetypal examples.


Let TT be the Lawvere theory for rings, viz. the category opposite to the category of finitely generated free rings (which are non-commutative polynomial rings X 1,,X n\mathbb{Z}\langle X_1, \ldots, X_n\rangle) and ring maps between them. Then for CC a category with finite products, a ring object in CC may be identified with a product-preserving functor TCT \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 RR in CC together with morphisms a:R×RRa: R \times R \to R (addition), m:R×RRm: R \times R \to R (multiplication), 0:1R0: 1 \to R (zero), e:1Re: 1 \to R (multiplicative identity), :RR-: R \to R (additive inversion), subject to commutative diagrams in CC that express the usual ring axioms.


Revised on August 21, 2015 02:58:04 by Urs Schreiber (