ring object (changes) in nLab

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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

Traditional definition

The traditional definition, based on a traditional presentation of the equational theory of rings, is that a ring object consists of an object RR in a cartesian monoidal category CC together with morphisms a:R×R→Ra: R \times R \to R (addition), m:R×R→Rm: R \times R \to R (multiplication), 0:1→R0: 1 \to R (zero), e:1→Re: 1 \to R (multiplicative identity), −:R→R-: R \to R (additive inversion), subject to commutative diagrams in CC that express the usual ring axioms.

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 ℤ⟨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 T→CT \to C.

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 rig objects internal to anybimonoidal category (a pseudomonoid in (SymMonCats,⊗ 𝔽,𝔽)(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})).

Namely, a semiring rig in abimonoidal 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; rig;
• A morphism

  μ R +:R⊕ 𝒞R⟶R\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⊗ 𝒞R⟶R\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 𝒞⟶R\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 𝒞⟶R\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 $\mathcal{C}$;

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

3. The diagrams

   \begin{imagefromfile} “file_name”: “semiring_in_bimonoidal_bilinearity_1.png”, “width”: 800 \end{imagefromfile}

   corresponding to the semiring rig axiomsa(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

   \begin{imagefromfile} “file_name”: “semiring_in_bimonoidal_bilinearity_2_corr.png”, “width”: 500 \end{imagefromfile}

   corresponding to the semiring rig axioms0a=00a=0 and a0=0a0=0 commute;

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

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

Examples

• A ring object in Top is a topological ring.

• A ring object in Ho(Top) is an H-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.

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

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

• monoid, monoid object,

• group, group object, group object in an (∞,1)-category

• ring, ring object

References

• Morten Brun, Witt Vectors and Equivariant Ring Spectra, 2006. Proceedings of the London Mathematical Society, Volume 94, pp. 351–385. (arXiv:math/0411567, doi:10.1112/plms/pdl010.)