Timeline for Is there a good general definition of "sheaves with values in a category"?
Current License: CC BY-SA 4.0
8 events
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| Apr 6, 2021 at 2:56 | comment | added | Peter LeFanu Lumsdaine | @ZhenLin: Sure, calling it “intensional” makes it sound like something scary from logic. But if instead you say “this notion of $A$-valued sheaves requires extra structure on $A$, and can depend on the choice of this structure”, I don’t think most mathematicians will be scared at all — that’s a very familiar phenomenon when generalising. | |
| Apr 6, 2021 at 0:57 | comment | added | Mike Shulman | Maybe this is a good way to motivate the ordinary mathematician to be less averse to logic. (-: | |
| Apr 5, 2021 at 23:51 | comment | added | Zhen Lin | Curiosity and, to be honest, a wish for a good general definition that could be presented to "ordinary" mathematicians who seem to be averse to "logic". (This is perhaps stretching the notion of "ordinary" a bit.) The fact that the naïve definition is repeated in many places without any caveats is quite frustrating for me but I wonder if that is because no alternative has appeared. | |
| Apr 5, 2021 at 23:35 | comment | added | Mike Shulman | @ZhenLin What motivates you to look for an extensional approach? | |
| Apr 5, 2021 at 23:34 | comment | added | Mike Shulman | @SimonHenry Indeed! I believe this is an issue for local rings too -- you don't get the "local homomorphisms" from a classifying topos. | |
| Apr 5, 2021 at 22:41 | comment | added | Zhen Lin | I, too, am of the opinion that the correct definition is supplied by internal logic. The difficulty with this approach is that it is "intensional", in the sense that the X in "sheaves of X" needs to have an a priori definition in terms of logic (whether the traditional kind or the (co)limit sketch kind); what I am hoping for is an "extensional" approach where X can be a "black box" category. As you say, going from "intensional" to "extensional" is lossy, but perhaps there is a canonical "intension" for every "extension" that is maximal or minimal or otherwise universal in some sense. | |
| Apr 5, 2021 at 17:26 | comment | added | Simon Henry | It should be noted however that in the case where condition (4) & (6) are satisfied, it is for a notion of morphisms that is forced upon you by the definition of objects you used. For example, for finite sets (or finite group) it is going to be either surjection between finite set (using Kuratowski finitness) or bijection between finite sets only (using "cardinal finite") and there is no way to have finite sets/groups with all morphisms between them from a geometric theory. | |
| Apr 5, 2021 at 15:18 | history | answered | Mike Shulman | CC BY-SA 4.0 |