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sarahzrf (Apr 08 2020 at 16:40):anyway, i was wondering about the comonad induced on Sh(X_d) from the geometric morphism Sh(X_d) → Sh(X) (when X_d is the space X but with the discrete topology)
sarahzrf (Apr 08 2020 at 16:40):and i was hoping maybe i could compute inverse images using étalé spaces >.>
sarahzrf (Apr 08 2020 at 16:40):but uh, maybe that's not so nice in this case.
sarahzrf (Apr 08 2020 at 16:41):to be precise, i was wondering if maybe you can recover the interior operator as an endomorphism of Ω or something
sarahzrf (Apr 08 2020 at 16:41):like maybe the comonad does something something interiors and you can equip Ω as a coalgebra
sarahzrf (Apr 08 2020 at 16:42):but hmm maybe interiors are not so relevant actually when i look at wikipedia's more explicit formula for inverse image >.<
sarahzrf (Apr 08 2020 at 16:43):the kan extension lined up the wrong way...
Morgan Rogers (he/him) (Apr 08 2020 at 16:45):Does looking at Lawvere-Tierney topologies help?
sarahzrf (Apr 08 2020 at 16:45):good suggestion! i dont know!
sarahzrf (Apr 08 2020 at 16:45):i have dabbled with those, but i dont have a strong feel for how everything connects
sarahzrf (Apr 08 2020 at 16:46):but oen thing i've noticed is that lawvere-tierney topologies are about closures rather than interiors :thinking:
sarahzrf (Apr 08 2020 at 16:46):which, once it finally registered to me, seemed bizarre given that we work with open sets
Morgan Rogers (he/him) (Apr 08 2020 at 16:46):Weird, I typed exactly that, then started daydreaming, and it vanished
sarahzrf (Apr 08 2020 at 16:46):heh
sarahzrf (Apr 08 2020 at 16:47):i think i started noticing, especially after some of john's bi-heyting stuff, that like...
sarahzrf (Apr 08 2020 at 16:47):there seems to be some odd stuff going on with closures and open vs closed and stuff, that i don't quite have my finger on...
sarahzrf (Apr 08 2020 at 16:48):e.g., ¬¬ is a closure operator! and open sets form a heyting algebra! and then i immediately jump to "aha! ¬¬ is topological closure!"
but of course it's not, because it stays within the algebra
sarahzrf (Apr 08 2020 at 16:48):it's like, "regularization"—interior of the closure
sarahzrf (Apr 08 2020 at 16:49):but if you try to use, say, the borel sets or something, so that you have both opens and closeds involved, then ¬ just becomes an involution, you have a boolean algebra—at least, assuming a classical metatheory
sarahzrf (Apr 08 2020 at 16:50):v mysterious
sarahzrf (Apr 08 2020 at 16:50):someone tell me what's going on
Morgan Rogers (he/him) (Apr 08 2020 at 16:52):It's a clash of terminology: "closure" describes any idempotent order-increasing endomorphism, which one happens to have for subsets of topological spaces thanks to closed sets being...closed...under arbitrary intersections. Read: "we use the verb 'to close' far too much"
Morgan Rogers (he/him) (Apr 08 2020 at 16:54):Especially since there are "closed subtoposes" (which do carry the geometric intuition of corresponding to closed subspaces in the case of sheaves on a space), which means that the corresponding Lawvere Tierney topologies are "closed closure operators" :face_palm: And let's not even get started on the abuse of the word "topology" in this context
sarahzrf (Apr 08 2020 at 16:55):i mean, i know re: terminology clash, but Things This Close In Concept Space Should Not Be Unrelated™
sarahzrf (Apr 08 2020 at 16:56):also, a lawvere-tierney topology is basically putting a compatible topology on every object of the topos at once, so i think it's a reasonable name :)
sarahzrf (Apr 08 2020 at 16:56):but a talk is starting so bbl
Morgan Rogers (he/him) (Apr 08 2020 at 16:57):Meh, sorry for over-explaining, then :joy:
sarahzrf said:
anyway, i was wondering about the comonad induced on Sh(X_d) from the geometric morphism Sh(X_d) → Sh(X) (when X_d is the space X but with the discrete topology)
Anyway, there should be a better answer to this^, I just don't have it on the top of my head rn.