# 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

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

Revised on April 24, 2014 10:24:08 by Adeel Khan (82.127.204.195)