My question is the following: is there an analog of [Morita theorem](http://ncatlab.org/nlab/show/Morita%20equivalence#classical_morita_theorem_11) in the simplicial setting?


I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories of simplicial modules $Mod(A)$ and $Mod(B)$ are equivalent (or maybe equivalent as simplicial categories?). Is there a result similar to the classical Morita theorem which gives a (simple) criterion for two rings to be Morita equivalent?


I tried to google, but I couldn't find anything useful. So I think maybe such an extension is just some trivial formality? Maybe it can easily be obtained from some well-known general result?


I have asked this [question](https://math.stackexchange.com/questions/567512/morita-theory-for-simplicial-rings) on math.stackexchenge, but didn't receive an answer (though comments there suggest that something like that should be true).


Thank you very much!



Morita theorem for simplicial rings
Recall that if $A$ is a ring and $\mathcal{C}$ is a cocomplete $\mathsf{Ab}$-category, then cocontinuous $\mathsf{Ab}$-functors $F : \mathsf{Mod}(A) \to \mathcal{C}$ correspond to left $A$-module objects $(M,\theta)$ in $\mathcal{C}$ (i.e. $M \in \mathcal{C}$ and $\theta : A \to \mathrm{End}(M)$ is a ring homomorphism). You can either prove this directly (define $M=F(A)$ etc.), or using that $\mathsf{Mod}(A)$ is the free $\mathsf{Ab}$-enriched cocompletion of the one-object $\mathsf{Ab}$-category $A$. The Theorem of Eilenberg-Watts is the special case $\mathcal{C}=\mathsf{Mod}(B)$ for another ring $B$. The special case of equivalences of categories is the Morita Theorem.


But free cocompletions work for an arbitrary cosmos. For instance we can take the category of simplicial abelian groups $\mathsf{simpAb}$. It follows that if $A$ is a simplicial ring and $\mathcal{C}$ is a cocomplete $\mathsf{simpAb}$-category, then cocontinuous $\mathsf{simpAb}$-functors $\mathsf{Mod}(A) \to \mathcal{C}$ correspond to $A$-module objects in $\mathcal{C}$. We get Eilenberg-Watts and then also Morita Theorem for simplicial rings: $\mathsf{Mod}(A) \simeq \mathsf{Mod}(B)$ *as simplicial* categories iff there is a simplicial $(A,B)$-bimodule $M$ and a simplicial $(B,A)$-bimodule $N$ such that $M \otimes\_B N \cong A$ and $N \otimes\_A M \cong B$.


