# nLab simplicial ring

## Theorems

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

It 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.

There is an evident notion of (∞,1)-category of modules over a simplicial ring. The corresponding bifibration $sMod \to sRing$ of modules over simplicial ring is equivalently to the tangent (∞,1)-category of the (∞,1)-category of simplicial rings.

## Properties

### Homotopy groups

Given a simplicial ring $A_\bullet$, it is standard that its connected components (the 0th “homotopy group”) $\pi_0(A)$ is an ordinary ring and $\pi_n(A)$ is a module over it, so $A_\bullet$ is weakly contractible iff $\pi_0(A)=0$.

Also, $\pi_0(A)$ is just the cokernel of $d_0-d_1:A_1\to A_0$. We can take any chain of face maps $A_n\to A_0$ and compose with the projection to $\pi_0(A)$, and this will be independent of which chain we chose. These maps $A_n\to\pi_0(A)$ 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.

All simplicial fields are constant. This is just 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.

### 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

## References

An introduction is in chapter 4 of

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.

Revised on November 7, 2014 15:35:43 by Adeel Khan (77.182.65.125)