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I noticed only now that the entry bimodule is in bad shape and needs some attention. For the moment I have added here a mentioning of the 2-category of algebras, bimodules and intertwiners and a pointer to the Eilenberg-Watts theorem.
I quote from the current entry:
Remark 4.2. As this notation suggests, $BMod_R$ is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules. spring
Does anyone know what the ’spring’ is doing at the end or should it be deleted?
It’s a device Urs uses sometimes when editing, I think to remember where to begin again from after stopping. If it’s still there after some time, it can be deleted!
Added reference to module over a monad.
Are $R$-bimodules definable for non-associative unital integer algebras $R$? Current definitions only assume $R$ to be an associative unital integer algebra, but the definition of a Jordan superalgebra requires $R$ to not be associative.
There is a weird definition of tensor product of bimodules over rings in this entry. I think it’s not correct. (At least, I don’t think it’s the definition of the standard tensor product.)
Oof. Thanks for pointing this out, J-B.
On the other hand, what you wrote isn’t what we normally understand by the tensor product of bimodules, either. It should be $A \otimes_N B$. I’ll come back to correct in a while, if no one else has.
Oh, I wasn’t sure about that. Maybe you must define $A \otimes_{N} B$ as the coequalizer of $A \otimes \lambda^{B}, \rho^{A} \otimes B: A \otimes N \otimes B \rightarrow A \otimes B$?
Yes.
Re #10:
This seems to originate in revision 36 from May 2022, part of about a dozen of anonymous edits.
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