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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2014

    I have given external tensor product its own entry.

    What I would really like to do for the moment is record there sufficient conditions under which the fiber over X 1×X 2X_1 \times X_2 is generated from external tensor products. I have added two references that discuss this for quasicoherent sheaves, but otherwise there is no discussion yet. Am being interrupted now.

    (What I really want eventually is conditions such that Mod(X 1×X 2)Mod(X 1) Mod(*)Mod(X 2)Mod(X_1 \times X_2) \simeq Mod(X_1) \otimes_{Mod(\ast)}Mod(X_2) )

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2022

    added a textbook reference, for completeness:

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 15th 2022

    Corrected the authors: it’s three authors, not one:

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2022

    That’s what I used to think, but when I grabbed the reference yesterday and looked at its publisher page here I saw a cover showing only Greub as author – which made me wonder to myself that maybe only the later volumes are coauthored.

    I should have known better of course. And on another page here the publishing house lists three authors but chooses to heavily misspell Greub’s name. (Apparently these pages are all computer-generated with no editor looking over them.)

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 16th 2022

    (Apparently these pages are all computer-generated with no editor looking over them.)

    Yes, these are OCRed scans, so they also misspelled Halperin and the title of the book (“De Rbam”)!

    Also, in the table of contents Greub’s name is misspelled in one more way: Grab.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2022

    Yes, clearly they are not re-investing the revenue they make from exploiting academia. The joke is on us.

    I had been well aware of the general phenomenon, but that they now auto-generate false covers for their own publications did catch me off-guard.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023
    • (edited Apr 27th 2023)

    I have added here a proof that if the monoidal fibration satisfies enough of the motivic yoga, then the external tensor product preserves colimits in each variable.

    I’d like to conclude next that in the case of a pseudofunctor of the form sFunc(;C):sSetGrpd opCatsFunc(-;\mathbf{C}) \,\colon\, sSet Grpd^{op} \to Cat, for a locally presentable sSetsSet-enriched category C\mathbf{C}, the Grothendieck construction is locally presentable (since all its ingredients are and so by this Prop.) so that, by the adjoint functor theorem, the above implies an internal hom-functor right adjoint to the external tensor (the “external internal hom”!? or “internal external hom”?! :-)

    diff, v10, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023

    added statement and proof (here), that for an indexed monoidal category with strong closed pullback and indexed coproducts (i.e. Wirthmueller-style motivic yoga), we have not only

    (f×g) *(𝒱𝒲)(f *𝒱)(g *𝒲) (f \times g)^\ast (\mathscr{V} \boxtimes \mathscr{W}) \;\simeq\; \big(f^\ast \mathscr{V}\big) \boxtimes \big(g^\ast \mathscr{W}\big)

    but also

    (f×g) !(𝒱𝒲)(f !𝒱)(g !𝒲). (f \times g)_! (\mathscr{V} \boxtimes \mathscr{W}) \;\simeq\; \big(f_! \mathscr{V}\big) \boxtimes \big(g_! \mathscr{W}\big) \,.

    diff, v12, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023
    • (edited May 16th 2023)

    made explicit (here) the corollary that pull/push-adjuncts of external tensor products of morphisms are given by external tensor product of the separate adjuncts:

    ϕγ˜ϕ˜γ˜. \widetilde{ \phi \boxtimes \gamma } \;\simeq\; \widetilde \phi \boxtimes \widetilde \gamma \,.

    diff, v13, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2023

    In the example section (here) I have

    added some text to the example of external tensor of vector bundles

    added the example of external cartesian products in Cartesian Grothendieck constructions (essentially a copy of the example which I had just added there)

    diff, v14, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2023

    fixed a typo in the proof (from #7)

    diff, v15, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2023

    added (here) statement and proof of a formula for “external pushout-products”

    diff, v16, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2023

    added more pointers to the literature after the proposition here

    diff, v22, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2023

    added references on external tensor product of group representations (but I fail to find any “original” references on this notion)

    and on external smash product of retractive spaces and parameterized spectra

    diff, v22, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2023

    added pointer to:

    for the external product on cobordism rings

    diff, v24, current