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There is an equivalence between modules over rings and Beck modules (Abelian group objects in a slice category)

$$ \text{Mod}_R \cong \text{Ab}(\text{CRing}/R) $$

I think of this as similar to the relationship between presheafs and discrete fibrations.

$$ [C^{\text{op}}, \text{Set}] \cong \text{Dis}(C) $$

There is also an equivalence between profunctors and two sided discrete fibrations

$$ [C \times D^{\text{op}}, \text{Set}] \cong \text{Dis}(C, D) $$

I feel like there ought to be a similar kind of equivalence between bimodules and some sort of "discrete spans" or "Beck bimodule".

But I'm not really familiar with the language of modules over rings. I primarily learn about functors and type theory stuff.

I'm not sure this is the right way to put it. Something like the following doesn't quite feel right to me.

$$ \text{Mod}_{R \oplus S} \cong \text{Ab}(\text{CRing}/(R \oplus S)) $$

I'm not really sure if an equivalence between the 2-category of bimodules plus intertwiners and a 2-category of "discrete spans" or "Beck bimodule" makes sense.

  • Are you sure about the equivalence in your first sentence? We rather have ${\rm CMon}({\rm Mod}_R)\simeq {\rm CRing}/R$ where $\rm CMon$ stands for commutative monoid. For the noncommutative case ${\rm Mon}({}_R{\rm Mod}_R)\simeq {\rm Ring}/R$. – Berci Nov 09 '21 at 23:15
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    https://ncatlab.org/nlab/show/module#ModulesOverARingInTermsOfStabilizedSlices https://ncatlab.org/nlab/show/Beck+module#beck_modules_over_groups nlab seems to think so but I'm really not at all familiar with Beck module stuff. –  Nov 12 '21 at 03:40
  • A $(A,B)$-bimodule on an abelian group $M$ is equivalent to a $A\otimes B^{op}$-module structure on $M$ (where $B^{op}$ is the opposite ring to $B$). In particular, you can apply the first equivalence and deduce that the category of $(A,B)$-bimodules is equivalent to the abelian objects in $\mbox{CRing}/A\otimes B^{op}$. In particular, is the tensor product of $A$ and (the opposite of) $B$, not their direct sum $\oplus$. – Alessandro Feb 12 '25 at 13:41

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