Timeline for $\infty$-ary tensor product on a category
Current License: CC BY-SA 3.0
7 events
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| Jan 19, 2023 at 7:19 | comment | added | naahiv | Regarding Martin's third example and making a choice $A\to I$ natural in $A$: since, as you said, the $I\to I$ component should be an isomorphism, any $f,g\colon I\to A$ are necessarily equal by naturality, and thus $I$ is initial in $\mathcal{C}$. Therefore the construction works for semicocartesian categories (with a sequentially cocontinuous product). Just naming what you already wrote for future readers. | |
| Aug 21, 2013 at 22:50 | comment | added | fosco | I want to publicly thank you for answering to (part of) this related question mathoverflow.net/questions/138518/… | |
| Sep 12, 2012 at 9:05 | comment | added | Zhen Lin | I'm not sure how to resolve that problem. Perhaps zero should be excluded everywhere – leading to some kind of non-unital infinitary monoidal category – and then we adjoin the axioms for the unit from the theory for finitary monoidal categories. Just excluding partitions containing infinitely many copies of $0$ is troublesome for the statement of the associative law: if we do that, then $1 + 1 + 1 + \cdots = \omega$ and $0 + 1 = 1$ are both valid partitions, but the substituted partition $(0 + 1) + (0 + 1) + (0 + 1) + \cdots = \omega$ is not. | |
| Sep 12, 2012 at 7:40 | comment | added | Martin Brandenburg | Thank you Zhen. But this does not include the tensor product of modules, as you say. If $K$ is a field, then $\bigotimes_{i < \alpha} K$ is a twisted group algebra of the giant group ${K^*}^{\alpha} / {K^*}^{({\alpha})}$ (see the link in my question). So this contradicts the definition with the trivial partition $0 = \sum_{i < \alpha} 0$. How can we adjust the definition so that it works? For example, we may treat the zero case separatedly. | |
| Sep 8, 2012 at 20:07 | history | edited | Zhen Lin | CC BY-SA 3.0 |
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| Sep 8, 2012 at 19:23 | history | edited | Zhen Lin | CC BY-SA 3.0 |
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| Sep 8, 2012 at 19:17 | history | answered | Zhen Lin | CC BY-SA 3.0 |