nLab simplicial ring

Idea

A simplicial ring is a simplicial object in the category Ring of rings.

This may be understood conceptually as follows:

• as ordinary rings are algebras over the ordinary algebraic theory $T$ of rings, if we regard this as an (∞,1)-algebraic theory then simplicial rings model the $(\infty,1)$-algebras over that;

• the category Ring${}^{op}$ is naturally equipped with the structure of a pregeometry. The corresponding geometry (for structured (∞,1)-toposes) is $sRing^{op}$, the opposite of the category of simplicial rings.

Definition

A simplicial ring is a simplicial object in the category Ring of rings.

Properties

Connected components

Given a simplicial ring $A = A_{\bullet}$, its connected components (the 0th “homotopy group”)) is an ordinary ring

$$\pi_0(A) \in Ring \,.$$

Forming connected components is a functor from simplicial rings to plain rings, which is left adjoint to the inclusion of ordinary rings as simplicially constant simplicial rings, exhibiting a reflective subcategory inclusion

$$Ring \stackrel{\stackrel{\pi_0}{\longleftarrow}}{\hookrightarrow} Ring^{\Delta^{op}} \,.$$

(Toën 14, section 2.2)

So on formal duals of commutative (simplicial) rings this is a coreflection of affine schemes in affine derived schemes

$$CRing^{op} \stackrel{\hookrightarrow}{\longleftarrow} (CRing^{\Delta^{op}})^{op} \,.$$

Notice that coreflective embeddings are also given for instance by the inclusion of manifolds into formal manifolds. This is one way in which formal duals of simplicial rings manifest themselves as infinitesimal neighbourhoods of formal duals of plain rings.

Higher homotopy groups

The higher homotopy groups $\pi_n(A)$ of a simplicial ring $A_\bullet$ are naturally modules over the ring $\pi_0(A)$ of connected components, so $A_\bullet$ is weakly contractible, as a simplicial set, iff $\pi_0(A)=0$.

(Again this is a manifestation of the simplicial ring being just an infinitesimal thickening of its connected components.)

Notice that $\pi_0(A)$ is equivalently the cokernel of $d_0-d_1 \colon A_1 \longrightarrow A_0$. Accordingly, any chain of face maps $A_n \longrightarrow A_0$ compose with the projection to $\pi_0(A)$ is independent of the choices. These maps

$$A_n \longrightarrow \pi_0(A)$$

give a surjective map from $A_\bullet$ to the constant simplicial ring $\pi_0(A)$.

(This is just the simplest piece of the Postnikov tower.) If $\pi_0(A)\neq 0$ we can then compose with a surjective map to a constant simplicial field.

Model category structure

There is a model category structure on simplicial rings that presents $\infty$-rings. See model structure on simplicial T-algebras for more.

We describe here the model category presentation of the (∞,1)-category of modules over simplicial rings.

Let $A$ be a simplicial commutative algebra. Write $A SMod$ for the category which, with $A$ regarded as a monoid in the category $SAb$ of abelian simplicial groups is just the category of $A$-modules in $SAb$. This means that

• objects are abelian simplicial groups $N$ equipped with an action morphism $A \otimes N \to N$ of simplicial abelian groups;

Equip $A SMod$ with the structure of a model category by setting:

• a morphism $N_1\to N_2$ of $A$-modules is a weak equivalence resp. a fibration precisely if the underlying morphism of simplicial sets is a weak equivalence, resp. fibration, in the standard model structure on simplicial sets.

Proposition This defines a model category structure which is

• cofibrantly generated;

• proper;

• cellular.

Examples

Simplicial fields

All simplicial fields are simplicially constant. This is because the composite $A_0\xrightarrow{s_0^n}A_n\xrightarrow{d_0^n}A_0$ is the identity, so $d_0^n$ is surjective, but all field homomorphisms are injective, so $d_0^n$ is an isomorphism.

Related concepts

• simplicial algebra

• model structure on simplicial T-algebras

References

Introduction and survey includes

• Bertrand Toën, chapter 4 of Simplicial presheaves and derived algebraic geometry , lecture at Simplicial methods in higher categories (pdf)

• Bertrand Toën, Derived Algebraic Geometry (arXiv:1401.1044)

See model structure on simplicial algebras for references on the model structure discussed above.

Some of the above material is taken from this MO entry.

Discussion in the context of homotopy theory, hence for simplicial ring spectra includes

• Paul Goerss, Michael Hopkins, Simplicial Structured Ring Spectra (1999) (web)