Timeline for What is known about module categories over general monoidal categories?
Current License: CC BY-SA 4.0
6 events
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| Nov 19, 2021 at 0:27 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
updated links and paper details
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| Jul 27, 2017 at 19:57 | comment | added | Chris Schommer-Pries | The "module categories are categories of modules" yoga is probably very special to the finite rigid setting. We tried to be careful about what exactly was needed for this here:arxiv:1406.4204. An interesting example is the following (non-rigid) tensor category C. It is finite, semisimple and has two simple objects 1 and x. The object 1 is the unit and we have $x \otimes x = 0$. This determines the monoidal structure by linearity. If I remember right, the subcategory generated by x is a module category which is not of the form A-mod(C) for an algebra A in C. | |
| Jul 27, 2017 at 19:47 | comment | added | Chris Schommer-Pries | The Deligne tensor product doesn't exist for arbitrary abelian categories. But if you relax "abelian" to instead mean "has finite colimits" then there is a very general tensor product that always exists called the "Kelly tensor product". A similar statement for locally presentable categories is in Cor 2.2.5 of (arXiv:1105.3104). I think the categorical machinery underlying these results can be adapted to the case of tensor products of module cats in the generality you are considering, but I don't know of a reference that does it. | |
| Nov 25, 2009 at 18:09 | comment | added | Evan Jenkins | An Azumaya algebra is one in which the tensor actions $X \mapsto A \otimes X$ and $X \mapsto X \otimes A$ are equivalences between $\mathcal{C}$ and the categories of left $A \otimes A^{\text{op}}$-modules and right $A^{\text{op}} \otimes A$-modules, respectively. This definition comes from the paper by Van Oystaeyen and Zhang, "The Brauer Group of a Braided Monoidal Category." | |
| Nov 25, 2009 at 7:23 | comment | added | Reid Barton | Could you recall the definition of Azumaya algebra in a general braided monoidal category? | |
| Nov 25, 2009 at 7:08 | history | asked | Evan Jenkins | CC BY-SA 2.5 |