symmetric monoidal (∞,1)-category of spectra
Under some reasonable assumptions on that allow one to construct a (symmetric) monoidal tensor product on the category of abelian group objects internal to , a ring object can also be defined as a monoid object internal to that monoidal category .
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 -refinement of a multiplicative structure on a corresponding cohomology theory, with various forms of K-theory providing archetypal examples.
Let be the Lawvere theory for rings, viz. the category opposite to the category of finitely generated free rings (which are non-commutative polynomial rings ) and ring maps between them. Then for a category with finite products, a ring object in may be identified with a product-preserving functor .
The more traditional definition, based on a traditional presentation of the equational theory? of rings, is that a ring object consists of an object in together with morphisms (addition), (multiplication), (zero), (multiplicative identity), (additive inversion), subject to commutative diagrams in that express the usual ring axioms.