Timeline for Relationship between coherent toposes/coherent logic and coherent sheaves
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27, 2015 at 21:02 | comment | added | dorebell | Would someone like to organize these thoughts as an answer? I'm looking for a historical/etymological connection. My definition of a coherent topos is that it is a topos of sheaves on a coherent category, and this isn't obviously related to a coherent object in an abelian category (whose definition is clearly inspired by coherent sheaves) | |
| Mar 25, 2015 at 17:14 | comment | added | Todd Trimble | @user74230 Okay, thanks for your explanations. I'm not claiming expertise myself and I don't have Johnstone's Elephant to hand. I trust that SGA4 is the original source of "coherent topos", but since then categorical logicians have fleshed out the notion of "coherent logic" which is a restriction of geometric logic, and this I suspect is where the OP's concerns actually lie. In any case, I do think it's premature just to say point-blank "the claim is wrong and there is no connection" -- there is more to the story than has been reported thus far. | |
| Mar 25, 2015 at 16:50 | comment | added | user74230 | @ToddTrimble: The notion of "coherent topos" was introduced by Grothendieck in SGA4, and I assumed that subsequent use of the word coherent in matters related to topoi stems from that. In the context of SGA4 it is qcqs schemes that provide the model case for the definition (and though undoubtedly inspired by the word used in sheaf theory, it has always seemed to me to be an unfortunate choice of terminology due to the reasons I allude to above). Anyway, I am just speculating based on what I have read, so I may be wrong about the history. | |
| Mar 25, 2015 at 16:46 | comment | added | user74230 | @JérômePoineau: Indeed, an unfortunate typo; thanks for noting it. | |
| Mar 25, 2015 at 15:34 | comment | added | Jérôme Poineau | @user74230: Oda should be Oka. | |
| Mar 25, 2015 at 14:41 | comment | added | Todd Trimble | @user74230 I think you misunderstood my comment. The claim of the first paragraph of the OP has essentially to do with etymology, and it was the category theorists / categorical logicians who adapted the terminology, to the best of my knowledge. You are of course absolutely correct about the last question of the OP, but I think he really wants to know what is the terminological connection, and this is what Zhen Lin is addressing. | |
| Mar 25, 2015 at 14:29 | comment | added | user74230 | @ToddTrimble: Every qcqs scheme gives rise to a coherent topos, but generally on such schemes the structure sheaf is not coherent and so one cannot readily produce any interesting examples of coherent sheaves of modules. Also, the structure sheaf on a complex manifold is coherent (in Serre's sense) by Oda's theorem but is not coherent in the sense of coherent objects of the category of sheaves of modules on the complex manifold. Although the definition of coherent object of an abelian category is modeled on the notion of coherent sheaf, there are subtle but important differences. | |
| Mar 25, 2015 at 13:54 | comment | added | Todd Trimble | @ZhenLin Would you consider adding your comment as an answer? I think it reflects the true situation far more accurately than the comment before yours. | |
| Mar 25, 2015 at 8:28 | comment | added | Zhen Lin | The definition of coherent topos is built on the definition of coherent object, which is essentially the same as the definition of coherent sheaf. | |
| Mar 25, 2015 at 4:58 | comment | added | user74230 | That claim is wrong and there is no connection (the category of coherent sheaves is not a topos). The correct analogue in algebraic geometry is that "coherent topos" is akin to the condition on a scheme that it be quasi-compact and quasi-separated, so this is just a rare case of poor terminology choice by Grothendieck (much as his use of the phrase "maximal point" to mean "generic point" is an unfortunate choice due to the fact that in the affine case maximal ideals correspond to closed points, sort of the opposite extreme from generic points). | |
| Mar 25, 2015 at 4:34 | review | First posts | |||
| Mar 25, 2015 at 5:41 | |||||
| Mar 25, 2015 at 4:32 | history | asked | dorebell | CC BY-SA 3.0 |