Timeline for When is the $(F_!,F^*)$ counit a natural isomorphism?
Current License: CC BY-SA 4.0
12 events
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| Jun 26, 2020 at 15:01 | history | edited | Matthieu Romagny | CC BY-SA 4.0 |
f --> F
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| Jun 29, 2010 at 20:00 | comment | added | BCnrd | David C., your last sentence is the sort of thing I was looking for. Even within algebraic geometry that is a reasonable condition, by the way. Anyway, the question as posed gave no context, so it sounded like random category-theoretic nonsense. You have explained why it is a reasonable question; thanks. | |
| Jun 29, 2010 at 18:59 | vote | accept | David Spivak | ||
| Jun 29, 2010 at 18:15 | comment | added | David Carchedi | @BCnrd: I disagree. You should keep in mind, although topoi may have been invented in algebraic geometry, they live and breathe outside of it. This counit being an isomorphism is equivalent to the induced geometric morphism being a connected morphism of topoi. These generalize maps of topological spaces which have connected fibers to the world of topoi. | |
| Jun 29, 2010 at 17:59 | answer | added | David Carchedi | timeline score: 5 | |
| Jun 29, 2010 at 3:30 | comment | added | BCnrd | The question sounds extremely unnatural (thinking about the case when $F$ corresponds to a map of sheaf categories associated to a morphism of schemes, say using etale topologies). Is there a reason to expect an interesting example to satisfy such a conclusion? Basically, what is the reason for posing this question? | |
| Jun 29, 2010 at 2:48 | answer | added | Tom Goodwillie | timeline score: 8 | |
| Jun 28, 2010 at 23:06 | comment | added | David Spivak | Tom: yes, yes, and yes (respectively). | |
| Jun 28, 2010 at 22:37 | comment | added | Tom Goodwillie | There is still a TeX problem, and also an unfortunate need to read between the lines to understand the question. Reading between the lines, $F^*$ is composition with $F$, "counit to" is "counit of", and $F_*$ and $F_!$ are respectively the right and the left adjoint of $F^*$, yes? | |
| Jun 28, 2010 at 22:29 | comment | added | Harry Gindi | Fixed a LaTeX problem. | |
| Jun 28, 2010 at 22:27 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 2 characters in body; added 10 characters in body; deleted 2 characters in body
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| Jun 28, 2010 at 21:24 | history | asked | David Spivak | CC BY-SA 2.5 |