Timeline for Examples of nonpointwise Kan extensions that "play a mathematical role"
Current License: CC BY-SA 4.0
7 events
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| Feb 10, 2022 at 22:50 | comment | added | Zhen Lin | I'm not familiar with derived functors used in practice. In general theory – model categories, derived categories of chain complexes with K-injective resolutions, deformable functors on homotopical categories, etc. – it seems to me that when derived functors exist for general reasons they are absolute Kan extensions. Maltsiniotis advocates taking absoluteness as part of the definition in order to get adjointness for free. | |
| Feb 10, 2022 at 18:41 | comment | added | varkor | @ZhenLin: that's an interesting question. I'm not familiar enough with derived functors to know whether this would be a reasonable example. If there are examples of derived functors that people care about that are not absolute/pointwise, then it seems like a good example. But if in practice, one could define derived functors in terms of absolute Kan extensions without losing motivating examples, then probably not. | |
| Feb 10, 2022 at 18:16 | comment | added | varkor | @TimCampion: I meant that we can construct Kan extensions specifically for the purpose of being examples of nonpointwise Kan extensions (one example being Exercise 3.9.7 of Borceux, though your examples probably also qualify in this sense), rather than being constructed for some useful purpose. | |
| Feb 10, 2022 at 14:27 | comment | added | Zhen Lin | Do you count derived functors, which are classically defined to be not necessarily pointwise but in practice not merely pointwise but even absolute? | |
| Feb 10, 2022 at 14:22 | comment | added | Tim Campion | Actually -- you say that it's straightforward to come up with examples of non-pointwise Kan extensions in general (when you don't ask for them to be "mathematically significant"). I find this claim interesting -- how do you do this? And what kinds of examples do you end up with? | |
| Feb 10, 2022 at 14:12 | comment | added | Tim Campion | I don't know many non-pointwise Kan extensions, and ones I do know don't seem to play any mathematical role. | |
| Feb 10, 2022 at 12:55 | history | asked | varkor | CC BY-SA 4.0 |