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nLab > Latest Changes: bimodule

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2013

    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.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeJun 25th 2020

    I quote from the current entry:

    Remark 4.2. As this notation suggests, BMod RBMod_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?

  3. 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!

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2020

    Right, sorry, thanks for catching. Have removed it now.

    diff, v29, current

    • CommentRowNumber5.
    • CommentAuthorvarkor
    • CommentTimeJul 27th 2022

    Added reference to module over a monad.

    diff, v41, current

    • CommentRowNumber6.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 14th 2022
    • (edited Dec 14th 2022)

    corrected typo. Sorry, I didn’t think that typos don’t require a comment.

    diff, v42, current

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeFeb 17th 2023

    Are RR-bimodules definable for non-associative unital integer algebras RR? Current definitions only assume RR to be an associative unital integer algebra, but the definition of a Jordan superalgebra requires RR to not be associative.

    • CommentRowNumber8.
    • CommentAuthorJ-B Vienney
    • CommentTime2 days ago

    Added definition of a bimodule over a monoid in a monoidal category

    diff, v43, current

    • CommentRowNumber9.
    • CommentAuthorJ-B Vienney
    • CommentTime2 days ago

    Added definition of tensor product of two bimodules over monoids in a monoidal category

    diff, v44, current

    • CommentRowNumber10.
    • CommentAuthorJ-B Vienney
    • CommentTime2 days ago
    • (edited 2 days ago)

    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.)

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTime2 days ago

    Oof. Thanks for pointing this out, J-B.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTime2 days ago

    On the other hand, what you wrote isn’t what we normally understand by the tensor product of bimodules, either. It should be A⊗ NBA \otimes_N B. I’ll come back to correct in a while, if no one else has.

    • CommentRowNumber13.
    • CommentAuthorJ-B Vienney
    • CommentTime2 days ago
    • (edited 2 days ago)

    Oh, I wasn’t sure about that. Maybe you must define A⊗ NBA \otimes_{N} B as the coequalizer of A⊗λ B,ρ A⊗B:A⊗N⊗B→A⊗BA \otimes \lambda^{B}, \rho^{A} \otimes B: A \otimes N \otimes B \rightarrow A \otimes B?

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTime2 days ago

    Yes.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTime1 day ago

    Re #10:

    This seems to originate in revision 36 from May 2022, part of about a dozen of anonymous edits.

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTime1 day ago

    More corrections under Tensor Product, and mention that monoids and bimodules between them form a bicategory, and that this construction can in turn be generalized to profunctors.

    diff, v48, current

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