Timeline for $\ell^1$ functor as left adjoint to unit ball functor
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jan 6, 2021 at 21:57 | comment | added | Yemon Choi | My misgivings about dividing by 2 were mainly aesthetic, since "taking the mean of two things" is not associative :) And yes I think it was in some place where I wanted to define or prove things about pullbacks where this annoyance arose | |
| Jan 6, 2021 at 21:41 | comment | added | Qiaochu Yuan | @Yemon: I'm not sure. We can talk about the entire Banach space of bounded maps, but singling out a particular bounded map (of norm $> 1$) requires talking about the "obvious" forgetful functor to $\text{Set}$ (not the unit ball functor); you don't actually get a morphism in the category so you can't use this sort of thing to e.g. reduce the computation of (co)equalizers to (co)kernels. It is a bit unsatisfying to have to divide by $2$ or whatever but on the other hand it's also pretty harmless. | |
| Jan 6, 2021 at 21:20 | comment | added | Yemon Choi | I used to get bothered by the fact that one often had to do ad hoc scaling tricks in constructions that would "naturally" be done using addition of morphisms in ${\rm Ban}_1$, but I guess your final comment says that this can and should be done using the closed structure -- would that be a fair interpretation? | |
| Jan 6, 2021 at 21:12 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
added 319 characters in body
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| Jan 6, 2021 at 21:04 | history | answered | Qiaochu Yuan | CC BY-SA 4.0 |