Timeline for Intuition for density comonad in relation to lifting problems
Current License: CC BY-SA 3.0
11 events
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| May 4, 2016 at 11:44 | vote | accept | Arrow | ||
| May 4, 2016 at 11:19 | comment | added | Arrow | @TomLeinster dualizing page 5 of these slides of yours, loosely speaking, the density comonad is what the composite of a functor with its right adjoint would be. I also know the density comonad is identity iff the functor is dense, so should (co)density be thought of as a condition that makes adjunction a literally invertible process? | |
| May 3, 2016 at 23:13 | answer | added | Omar Antolín-Camarena | timeline score: 6 | |
| May 3, 2016 at 19:33 | comment | added | Tom Leinster | Or maybe I can. Here's how to think about the density comonad of a functor $F$: it's what the comonad induced by $F$ and its right adjoint would be if $F$ had a right adjoint - but it's defined in many situations where $F$ doesn't have a right adjoint. | |
| May 3, 2016 at 19:31 | comment | added | Tom Leinster | If you'd like a general introduction to codensity monads, there's this: golem.ph.utexas.edu/category/2012/09/… . If you'd like an introduction to density comonads, I'm afraid I can't help :-) | |
| May 3, 2016 at 8:19 | comment | added | Arrow | @OmarAntolín-Camarena sorry for the late reply. I would like some help in understanding what the density comonad does. Conceptually, why does its counit describe generic lifting problems, and what does it do for more interesting $\mathsf J$? What does it measure about the functor $I$? | |
| May 3, 2016 at 8:18 | history | edited | Arrow | CC BY-SA 3.0 |
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| May 1, 2016 at 18:58 | comment | added | Omar Antolín-Camarena | In your Remark 2, what "second sentence" are you referring to? | |
| May 1, 2016 at 18:57 | comment | added | Omar Antolín-Camarena | I'm having a hard time understanding exactly what you are asking. Is part of the question what density comonads have to do with lifting problems? If so, maybe it helps to say that when $\mathsf{J}$ is discrete, the component at $f$ of the counit of $M$ is exactly the generic lifting problem (12.2.3). (But I get the feeling you understood that and wanted something else?) | |
| May 1, 2016 at 9:19 | history | edited | Arrow | CC BY-SA 3.0 |
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| Apr 30, 2016 at 14:03 | history | asked | Arrow | CC BY-SA 3.0 |