# 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 $\left(\infty ,1\right)$-algebras over that;

• the category Ring${}^{\mathrm{op}}$ is naturally equipped with the structure of a pregeometry. The corresponding geometry (for structured (∞,1)-toposes) is ${\mathrm{sRing}}^{\mathrm{op}}$, 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 $\mathrm{sMod}\to \mathrm{sRing}$ 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 ${A}_{•}$, it is standard that its connected components (the 0th “homotopy group”) ${\pi }_{0}\left(A\right)$ is an ordinary ring and ${\pi }_{n}\left(A\right)$ is a module over it, so ${A}_{•}$ is weakly contractible iff ${\pi }_{0}\left(A\right)=0$.

Also, ${\pi }_{0}\left(A\right)$ 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 ${\pi }_{0}\left(A\right)$, and this will be independent of which chain we chose. These maps ${A}_{n}\to {\pi }_{0}\left(A\right)$ give a surjective map from ${A}_{•}$ to the constant simplicial ring ${\pi }_{0}\left(A\right)$.

(This is just the simplest piece of the Postnikov tower.) If ${\pi }_{0}\left(A\right)\ne 0$ we can then compose with a surjective map to a constant simplicial field.

All simplicial fields are constant. This is just because the composite ${A}_{0}\stackrel{{s}_{0}^{n}}{\to }{A}_{n}\stackrel{{d}_{0}^{n}}{\to }{A}_{0}$ 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 $A\mathrm{SMod}$ for the category which, with $A$ regarded as a monoid in the category $\mathrm{SAb}$ of abelian simplicial groups is just the category of $A$-modules in $\mathrm{SAb}$. This means that

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

Equip $A\mathrm{SMod}$ 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 December 29, 2012 17:15:27 by Urs Schreiber (89.204.155.26)