symmetric monoidal (∞,1)-category of spectra
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.
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.
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.
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
Equip $A SMod$ with the structure of a model category by setting:
Proposition This defines a model category structure which is
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.