nLab
ring object

Contents

Contents

Idea

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 “xyzxyz-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.

Definition

As a model of a Lawvere theory

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.

Via the microcosm principle

Alternatively, one may define ring objects following the Baez–Dolan microcosm principle. Indeed, similarly to how it is possible to define monoids in a monoidal category (a pseudomonoid in (Cat,×,pt)(\mathsf{Cat},\times,\mathsf{pt})), it is possible to speak of semiring objects internal to any bimonoidal category (a pseudomonoid in (SymMonCats, 𝔽,𝔽)(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})).

Namely, a semiring in a bimonoidal category (𝒞, 𝒞, 𝒞,0 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}},\oplus_{\mathcal{C}},\mathbf{0}_{\mathcal{C}},\mathbf{1}_{\mathcal{C}}) is given by a quintuple (R,μ R +,η R +,μ R ×,η R ×)(R,\mu^{+}_{R},\eta^{+}_{R},\mu^{\times}_{R},\eta^{\times}_{R}) consisting of

  • An object RR of 𝒞\mathcal{C}, called the underlying object of the semiring;
  • A morphism
    μ R +:R 𝒞RR\mu^{+}_{R}\colon R\oplus_{\mathcal{C}}R\longrightarrow R

    of 𝒞\mathcal{C}, called the addition morphism of RR;

  • A morphism
    μ R ×:R 𝒞RR\mu^{\times}_{R}\colon R\otimes_{\mathcal{C}}R\longrightarrow R

    of 𝒞\mathcal{C}, called the multiplication morphism of RR;

  • A morphism
    η R +:0 𝒞R\eta^{+}_{R}\colon\mathbf{0}_{\mathcal{C}}\longrightarrow R

    of 𝒞\mathcal{C}, called the additive unit morphism of RR;

  • A morphism
    η R ×:1 𝒞R\eta^{\times}_{R}\colon\mathbf{1}_{\mathcal{C}}\longrightarrow R

    of 𝒞\mathcal{C}, called the multiplicative unit morphism of RR;

satisfying the following conditions:

  1. The triple (R,μ R +,η R +)(R,\mu^{+}_R,\eta^{+}_R) is a commutative monoid in 𝒞 \mathcal{C} ;

  2. The triple (R,μ R ×,η R ×)(R,\mu^{\times}_R,\eta^{\times}_R) is a monoid in 𝒞 \mathcal{C} ;

  3. The diagrams

    corresponding to the semiring axioms a(b+c)=ab+aca(b+c)=a b+a c and (a+b)c=ac+bc(a+b)c=a c+b c commute;

  4. The diagrams

    corresponding to the semiring axioms 0a=00a=0 and a0=0a0=0 commute;

Moreover, for 𝒞\mathcal{C} a braided bimonoidal category, one defines a commutative semiring in 𝒞\mathcal{C} to be a semiring in 𝒞\mathcal{C} whose multiplicative monoid structure is commutative.

A partial version of this definition first appeared in (Brun 2006, Definition 5.1).

Examples

For the notion of a semiring in a bimonoidal category defined via the microcosm principle, we have the following examples.

  • A semiring in (Sets,,×,,×)\left(\mathsf{Sets},\coprod,\times,\emptyset,\times\right) is a monoid.
  • A semiring in (CMon,, ,0,)\left(\mathsf{CMon},\oplus,\otimes_\mathbb{N},0,\mathbb{N}\right) is a semiring.
  • A semiring in (Ab,, ,0,)\left(\mathsf{Ab}, \oplus,\otimes_\mathbb{Z},0,\mathbb{Z}\right) is a ring.
  • A semiring in (Mod R,, R,0,R)\left(\mathsf{Mod}_R,\oplus,\otimes_R,0,R\right) is an associative algebra.
  • A semiring in (Cats,,×, cat,pt)\left(\mathsf{Cats},\coprod,\times,\emptyset_{\mathsf{cat}},\mathsf{pt}\right) is a strict monoidal category.

References

Last revised on September 3, 2021 at 03:06:34. See the history of this page for a list of all contributions to it.