My question is the following: is there an analog of Morita theorem 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 on math.stackexchenge, but didn't receive an answer (though comments there suggest that something like that should be true).

Thank you very much!

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    $\begingroup$ Maybe helpful: arxiv.org/abs/math/0310146 $\endgroup$ – Qiaochu Yuan Nov 19 '13 at 20:06
  • $\begingroup$ Speaking off the top of my head here: I would expect the usual proof to be constructive, therefore you could reuse it in the internal language of simplicial sets. But one would have to check this. $\endgroup$ – Andrej Bauer Nov 20 '13 at 8:05

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 $N \otimes_A M \cong B$.

  • $\begingroup$ Dear Martin, thank you very much for your answer! Can you, please, explain to me, why the Eilenberg-Watts theorem holds in the latter case? I have found a paper by Hovey "The Eilenberg-Watts theorem in homotopical algebra", and there he states the theorem for any cosmos $M$, but only for "nice" functors. For example, there should be a natural map $K\otimes FX\to F(K\otimes X)$ for any $K\in M$, $X\in Mod(A)$. Can you, please, explain, why is it the case when $M=simpAb$ and $C=Mod(B)$? $\endgroup$ – Sasha Patotski Nov 24 '13 at 22:54
  • $\begingroup$ This corresponds to the enrichment. And I restrict to simplicial functors. $\endgroup$ – Martin Brandenburg Nov 26 '13 at 21:08

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