Timeline for Can we make Pres *-autonomous?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 7, 2020 at 22:04 | comment | added | Ivan Di Liberti | You are just right, I'll be thinking about this. One could try to fix what I said in a technical way (choosing only left adjoints, choosing total categories instead of complete and cocomplete ones), but I will try to come back with a more mature observation before shooting in the sky. | |
| Jun 7, 2020 at 21:57 | comment | added | Simon Henry | Is it ? I don't think the tensor product of complete & co-complete categories is defined. Also if you want 'op' to be a contravariant functor you need morphisms to have right adjoint. | |
| Jun 7, 2020 at 21:46 | comment | added | Ivan Di Liberti | Let me rephrase my previous comment. There exists a very natural fully faithful embedding of $\textbf{Pres}$ in a $*$-autonomous category preserving the monoidal closed structure. That is the inclusion of $\textbf{Pres}$ in the (2-)category of complete and cocomplete categories and cocontinuous functors. | |
| Jun 7, 2020 at 21:10 | comment | added | Ivan Di Liberti | Ahah, you are somehow missing my point. The 2-category of complete and cocomplete categories (and cocontinuous functors) is $*$-autonomous replacing adjoints with (co)continuous functors. The existence of a generator in a loc. pres category just makes the whole situation tamer. In a way, I am claiming that you already have the answer, you just care too much about (co)generating sets and smallness assumptions. From my point of view, we agree to agree. | |
| Jun 7, 2020 at 21:05 | comment | added | Simon Henry | Ok, let say that what I really mean is that the $0$-categorical analogue of $\mathbf{Pres}$ is $\mathbf{Sup}$, which I think you will agree with. I could argue that the $0$-categorical analogue of the category $\mathbf{CoComp}$ of co-complete categories is really the category $\mathbf{SUP}$ of possibly large posets having supremum of small families, which would contradict what you say. Though these matters are somehow subjective, so I guess we can agree to disagree ^^ | |
| Jun 7, 2020 at 20:59 | comment | added | Ivan Di Liberti | The fact that loc. pres. categories are complete is just a phenomenon of the more general pattern that every total category is complete. The generator forces the limits to exist computing them as "sups of lower bounds". | |
| Jun 7, 2020 at 20:53 | comment | added | Ivan Di Liberti | I disagree with you, Simon. The analogue of Sup is the 2-category of cocomplete categories. It happens that a cocomplete poset is complete, thus the opposite of a suplattice is a suplattice. For categories, you could maintain the analogy by looking at complete and cocomplete categories. Notice that even in Cat a cocomplete category is always very close to be complete (Adamek et al: Cocompleteness almost implies completeness). | |
| Jun 7, 2020 at 17:01 | history | asked | Simon Henry | CC BY-SA 4.0 |