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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 $X_1 \times X_2$ 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 $Mod(X_1 \times X_2) \simeq Mod(X_1) \otimes_{Mod(\ast)}Mod(X_2)$)
added a textbook reference, for completeness:
Corrected the authors: it’s three authors, not one:
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.)
(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.
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.
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 $sFunc(-;\mathbf{C}) \,\colon\, sSet Grpd^{op} \to Cat$, for a locally presentable $sSet$-enriched category $\mathbf{C}$, 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”?! :-)
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
$(f \times g)^\ast (\mathscr{V} \boxtimes \mathscr{W}) \;\simeq\; \big(f^\ast \mathscr{V}\big) \boxtimes \big(g^\ast \mathscr{W}\big)$but also
$(f \times g)_! (\mathscr{V} \boxtimes \mathscr{W}) \;\simeq\; \big(f_! \mathscr{V}\big) \boxtimes \big(g_! \mathscr{W}\big) \,.$added pointer to:
for the external product on cobordism rings
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