Timeline for Characterizing Isbell self-dual objects
Current License: CC BY-SA 3.0
8 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 23, 2017 at 9:08 | vote | accept | fosco | ||
| Jan 24, 2017 at 15:59 | comment | added | fosco | Argh! So $\lnot$"everything is a functor"? That's something I must remember next time I brag about the good old saying. Let's say that I want to know if, in any sense whatsoever, sending a category to "its" Isbell adjunction is functorial. | |
| Jan 24, 2017 at 15:02 | comment | added | Todd Trimble | By the way, there's one fact of life which is either disconcerting or interesting depending on your point of view, given by the saying "Injective Hulls are not natural", a title of a paper by Adamek, Herrlich, and Rosicky. An example is the MacNeille completion of a poset, which is a special case of the Isbell completion/envelope. So if you're hoping that Isbell completions are cleanly functorial, you're probably in for a disappointment. :-( I've been burned by this type of hope in the past. | |
| Jan 24, 2017 at 14:56 | comment | added | Todd Trimble | Maybe it depends what is meant by $Adj$ (a bicategory), and it may depend on which functor you mean. If morphisms in $Adj$ are pairs of functors which commute with both the left and right adjoint parts, then one thing to look at is a functor $Cat^{op} \to Adj$ that takes $f: A \to B$ to the pair $(V^{f^{op}}, (V^f)^{op})$, but a calculation shows this doesn't work. I didn't check, but I'm skeptical that other possibilities involving left or right adjoints to $V^{f^{op}}$ and $(V^f)^{op}$ work either. | |
| Jan 23, 2017 at 20:09 | comment | added | fosco | I have an additional question, is $A\mapsto \text{Spec}_A \dashv {\cal O}_A$ a functor from $Cat$ to the category $Adj$ of adjunctions? | |
| Jan 23, 2017 at 15:46 | comment | added | fosco | "I may come back and add more" you already helped a lot, but please come back! | |
| Jan 23, 2017 at 15:26 | history | answered | Todd Trimble | CC BY-SA 3.0 |