Timeline for $\infty$-ary tensor product on a category
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2022 at 20:39 | answer | added | varkor | timeline score: 3 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 12, 2012 at 21:50 | comment | added | Michal R. Przybylek | One may expect a good notion of infinitary monoidal category by replacing $\mathit{fin}$ with a larger category with “similar” properties. As Mike suggested in his answer, two natural choices are $\mathit{Core}(\mathbf{Set})$ and $\mathit{Core}(\mathbf{Ord})$. I guess, to get a "finitely braided" infinitary monoidal category one has to extend the definition of a braid group via transfinite recursion and use the category of ordinals with associated braid groups. Similarly, one may obtain "finitely symmetric" infinitary monoidal category. | |
| Sep 12, 2012 at 21:49 | comment | added | Michal R. Przybylek | Perhaps the crucial thing is that there is no single concept of infinitary monoidal category. I think, the right approach to define infinitary monoidal categories is along Mike's line. If we take the grupoid of finite ordinals $\mathit{fin}$ then the Grothendieck construction over the functor $\mathbb{C}^{(-)} \colon \mathit{fin}^{op} \rightarrow \mathbf{Cat}$ gives a kind of the "free" symmetric (here permutations define symmetry) monoidal category on $\mathbb{C}$. Then monoidal categories are just algebras of the induced monad $\mathbb{C} \mapsto \int \mathbb{C}^{(-)}$. (cont) | |
| Sep 10, 2012 at 3:27 | answer | added | Mike Shulman | timeline score: 3 | |
| Sep 8, 2012 at 19:51 | comment | added | Mike Shulman | Construction (3) uses a transfinite composition, and so do the largest Hausdorff quotient, the associated sheaf, and colimits of algebras. Transfinite composition is of course a widely applicable technique. But are you claiming that those three other constructions are actually infinitary tensor products? | |
| Sep 8, 2012 at 19:17 | answer | added | Zhen Lin | timeline score: 16 | |
| Sep 8, 2012 at 16:25 | comment | added | Martin Brandenburg | @Zhen: Yes, that's right. So is this spelled out somewhere in that generality? Or can you do this in an answer? Thanks. | |
| Sep 8, 2012 at 15:40 | comment | added | Zhen Lin | There is a notion of "unbiased monoidal category" in which all finite monoidal products are defined at once. This can be found in Leinster's book, Higher operads, higher categories. This seems easy to generalise in comparison to the traditional one involving only a monoidal unit and a binary monoidal product. The coherence axioms are based on partitions, as Buschi Sergio indicated. | |
| Sep 8, 2012 at 14:58 | comment | added | Buschi Sergio | A first idea is generalizing the associativity axioms to (finite) partitions on a set of (ordered) indexes, classically you have the finite set {1, 2, 3, 4} indexes and associate respect to partitions of it... | |
| Sep 8, 2012 at 14:45 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |