Timeline for Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 13, 2018 at 21:36 | review | Suggested edits | |||
| Oct 13, 2018 at 22:11 | |||||
| Oct 13, 2018 at 20:41 | answer | added | Noah Snyder | timeline score: 8 | |
| Oct 13, 2018 at 13:01 | vote | accept | Ben Webster♦ | ||
| Oct 13, 2018 at 12:48 | answer | added | Todd Trimble | timeline score: 11 | |
| Oct 13, 2018 at 11:40 | comment | added | Todd Trimble | Very related to what @მამუკაჯიბლაძე was saying, I might call this the "cocomplete tensor product". This works in general for enriched category over a complete cocomplete symmetric monoidal category $V$, but taking $V = Ab$ as here, for $C, D$ two ($V$-)cocomplete categories there is a cocomplete $C \boxtimes D$ and a functor $C \times D \to C \boxtimes D$ that is separately cocontinuous in its two arguments, that is the universal such: any separately cocontinuous $F: C \times D \to E$ to cocomplete $E$ extends uniquely (up to isomorphism) to a cocontinuous $\hat{F}: C \boxtimes D \to E$. | |
| Oct 13, 2018 at 7:45 | comment | added | YCor | If I understand correctly, you're asking not just for a name, but for a categorical construction (and its possible name, if it has been done somewhere) which when specified to $A$-mod and $B$-mod yields $A\otimes B$-mod (I guess $A,B$ are meant to be commutative algebras over a common commutative ring). | |
| Oct 13, 2018 at 4:34 | comment | added | მამუკა ჯიბლაძე | I believe abstractly this is $\operatorname{Adj}({\mathcal C}_1^{\mathrm{op}},{\mathcal C}_2)$, the category of contravariant adjoint pairs between these categories (an $A\otimes B$-module $M$ corresponds to the pair $\left\langle\operatorname{Hom}_A(-,M),\operatorname{Hom}_B(-,M)\right\rangle$, and any such pair $\left\langle L,R\right\rangle$ is isomorphic to one such, with $M\cong L(A)\cong R(B)$). I would call this the category of Galois connections but I never met either this or any other name for it. | |
| Oct 13, 2018 at 2:55 | comment | added | Dylan Wilson | If you use dualizable modules then the Deligne tensor product should work. I think there's a version for some kind of presentable categories but I'm not so familiar with the non-derived version; if you use derived categories then the tensor product of presentable infty-categories does this. | |
| Oct 13, 2018 at 2:31 | history | asked | Ben Webster♦ | CC BY-SA 4.0 |