Timeline for The shape of an ($\infty$-)topos as a monad
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jul 29 at 17:35 | comment | added | Marc Hoyois | I see the problem: it's not true that composition coincides with products of pro-objects. This is true if the second term of the composition preserves filtered colimits, but not in general. So cartesian monads should not be just pro-spaces. | |
| Jul 29 at 17:27 | comment | added | Marc Hoyois | Is the first question then equivalent to "is every pro-space the shape of a topos"? I'm confused because this fails even for pro-sets and 1-topoi. | |
| Jul 28 at 18:17 | comment | added | Tim Campion | Ah — the composition monoidal structure on $Fun^{lex,acc}(Spaces,Spaces)$ in fact coincides with the cocartesian monoidal structure. So every such functor admits a unique monad structure. This is also true in the 1-categorical setting, so Johnstone must be giving a description of arbitrary pro-sets. I wonder if he realized that… | |
| Jul 27 at 16:17 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jul 27 at 16:06 | history | asked | Tim Campion | CC BY-SA 4.0 |