Timeline for $\ast$-autonomous categories with non-invertible dualizing object?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jan 16 at 17:46 | comment | added | Mike Shulman | Well, if you quote a definition in your question saying that it's symmetric, you shouldn't be surprised if people only give you answers that are symmetric... (-: | |
| Jan 16 at 7:47 | comment | added | Max Demirdilek | @MikeShulman No, not in general. But in my book a $\ast$-autonomous category isn’t required to be symmetric monoidal. (I simply used the nLab-reference out of convenience.) | |
| Jan 16 at 2:16 | comment | added | Mike Shulman | @MaxDemirdilek Is that tensor product symmetric? | |
| Jan 15 at 22:11 | comment | added | Max Demirdilek | @MikeShulman Another contender is, for a finite-dimensional (associative, unital) algebra $A$ over a field $k$, the category of finite-dimensional $A$-bimodules. This becomes a monoidal category via the tensor product over $A$. The linear dual $A^*$ is a dualizing object. | |
| Oct 15, 2023 at 17:50 | vote | accept | Max Demirdilek | ||
| Oct 14, 2023 at 1:56 | answer | added | Tim Campion | timeline score: 3 | |
| Sep 16, 2022 at 16:12 | comment | added | Mike Shulman | (I mean, other than categories that are compact closed and hence degenerately $\ast$-autonomous. The other contender is suplattices, of course.) | |
| Sep 16, 2022 at 16:08 | comment | added | Mike Shulman | @ChrisH Cool! That might be the most "naturally-occurring" $\ast$-autonomous category I've ever encountered. It would be great to have that as an example on the nLab (hint, hint)... | |
| Sep 15, 2022 at 8:53 | comment | added | Chris H | It is! I don’t have a reference off hand, but one can prove it without much trouble using the six functor formalism and that duality respects external products. | |
| Sep 14, 2022 at 22:54 | comment | added | Mike Shulman | @ChrisH Is that category $\ast$-autonomous? I didn't know that. | |
| Sep 13, 2022 at 21:34 | comment | added | Chris H | The constructible category of sheaves on a singular variety might be an example, there doesn’t seem to be obvious dualising maps for the dualising sheaf. | |
| Sep 13, 2022 at 17:50 | history | edited | Max Demirdilek | CC BY-SA 4.0 |
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| Sep 13, 2022 at 17:19 | history | edited | Max Demirdilek | CC BY-SA 4.0 |
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| Sep 13, 2022 at 17:08 | history | edited | Max Demirdilek | CC BY-SA 4.0 |
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| Sep 13, 2022 at 17:02 | comment | added | Qiaochu Yuan | The OP is referring to the general notion of a dualizable object in a monoidal category (which is not the same as a dualizing object): ncatlab.org/nlab/show/dualizable+object | |
| Sep 13, 2022 at 16:48 | comment | added | Max New | What unit and counit are you asking to be invertible? | |
| Sep 13, 2022 at 16:23 | history | edited | Max Demirdilek |
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| Sep 13, 2022 at 16:11 | history | asked | Max Demirdilek | CC BY-SA 4.0 |