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!

  • 6
    $\begingroup$ Maybe helpful: arxiv.org/abs/math/0310146 $\endgroup$ Commented Nov 19, 2013 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. $\endgroup$ Commented Nov 20, 2013 at 8:05

1 Answer 1


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 $C=Mod(B)$? $\endgroup$ Commented Nov 24, 2013 at 22:54
  • $\begingroup$ This corresponds to the enrichment. And I restrict to simplicial functors. $\endgroup$ Commented Nov 26, 2013 at 21:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.