arthur
@arthur-i.bsky.social
having math withdrawal........
July 8, 2024 at 4:57 PM
having math withdrawal........
need to be locked back in to making art so no math for like a week or something 😔 I literally have to stop myself from looking at anything math-related lol it's so dire
July 4, 2024 at 7:42 PM
need to be locked back in to making art so no math for like a week or something 😔 I literally have to stop myself from looking at anything math-related lol it's so dire
need mathematicians to make CAT real at some point. I've written CAT//Set a few times now and I always feel guilty lol
July 3, 2024 at 5:31 PM
need mathematicians to make CAT real at some point. I've written CAT//Set a few times now and I always feel guilty lol
are you fucking kidding
July 3, 2024 at 10:39 AM
are you fucking kidding
I'm very annoyed that I can't just focus on both math and art at the same time lol, like learning about adjoint functors has been very fun and all but I need to be spending my time drawing hot men!!!!!
July 2, 2024 at 7:24 PM
I'm very annoyed that I can't just focus on both math and art at the same time lol, like learning about adjoint functors has been very fun and all but I need to be spending my time drawing hot men!!!!!
why are limits so bad lol, I've been trying to generalize (co)limit functors and finally figured out you can turn colimits into a functor Cat//C -> C, but for limits you have to use something disgusting like (Cat^co//C)^op -> C 🤮
July 1, 2024 at 4:03 PM
why are limits so bad lol, I've been trying to generalize (co)limit functors and finally figured out you can turn colimits into a functor Cat//C -> C, but for limits you have to use something disgusting like (Cat^co//C)^op -> C 🤮
I'm probably completely overthinking this but anyway, if F : Set -> Ab is free and G : Ab -> Set is forgetful, there's a natural isomorphism Ab(Z,-) -> Set(*,G(-)):
June 29, 2024 at 11:14 PM
I'm probably completely overthinking this but anyway, if F : Set -> Ab is free and G : Ab -> Set is forgetful, there's a natural isomorphism Ab(Z,-) -> Set(*,G(-)):
idk if this is like, a very standard way of doing it, but I just figured out you can very elegantly construct the free functor Set -> Ab as the Yoneda extension of the functor * -> Ab sending the single object to Z, since Psh(*) is isomorphic to Set. I'm so happy about this
June 27, 2024 at 5:31 PM
idk if this is like, a very standard way of doing it, but I just figured out you can very elegantly construct the free functor Set -> Ab as the Yoneda extension of the functor * -> Ab sending the single object to Z, since Psh(*) is isomorphic to Set. I'm so happy about this
I wish I could get back into cohomology but unfortunately I literally could not care less about cohomology classes 😔
June 19, 2024 at 12:08 PM
I wish I could get back into cohomology but unfortunately I literally could not care less about cohomology classes 😔
just learned about the nerve functor....... the nerve and realization adjunction is so fucked up but also kind of one of the most incredible things ever
June 13, 2024 at 4:17 PM
just learned about the nerve functor....... the nerve and realization adjunction is so fucked up but also kind of one of the most incredible things ever
love studying category theory without thinking about size issues 😇 (<- just wrote a limit diagram with Cat as one of the objects and foolishly looked up what the category of large categories is called)
June 11, 2024 at 9:52 AM
love studying category theory without thinking about size issues 😇 (<- just wrote a limit diagram with Cat as one of the objects and foolishly looked up what the category of large categories is called)
I've been so distracted lately lol I went from "yoneda extensions are insanely cool" to "I need comma categories to make sense formally or I will literally lose sleep over this for the rest of my life" to "oooh subobject classifier shiny"
June 9, 2024 at 12:58 AM
I've been so distracted lately lol I went from "yoneda extensions are insanely cool" to "I need comma categories to make sense formally or I will literally lose sleep over this for the rest of my life" to "oooh subobject classifier shiny"
just saw the definition of a comma category using a single pullback this is insane wtf, actually gave me goosebumps for a little bit which I did not expect to come from comma categories of all things lol
June 7, 2024 at 10:20 AM
just saw the definition of a comma category using a single pullback this is insane wtf, actually gave me goosebumps for a little bit which I did not expect to come from comma categories of all things lol
ok comma categories have been bothering me a bit
I think it's mainly the definition I'm working with, where you define objects to be morphisms and morphisms to be pairs of morphisms that make commuting squares. but my issue is you can have the same pair of morphisms create different squares:
I think it's mainly the definition I'm working with, where you define objects to be morphisms and morphisms to be pairs of morphisms that make commuting squares. but my issue is you can have the same pair of morphisms create different squares:
June 3, 2024 at 10:56 PM
ok comma categories have been bothering me a bit
I think it's mainly the definition I'm working with, where you define objects to be morphisms and morphisms to be pairs of morphisms that make commuting squares. but my issue is you can have the same pair of morphisms create different squares:
I think it's mainly the definition I'm working with, where you define objects to be morphisms and morphisms to be pairs of morphisms that make commuting squares. but my issue is you can have the same pair of morphisms create different squares:
for some reason I used to silently complain about category theory being aggressively functorial and natural, like the double naturality thing in the yoneda lemma or adjunctions, but I get it now......................
May 31, 2024 at 4:12 PM
for some reason I used to silently complain about category theory being aggressively functorial and natural, like the double naturality thing in the yoneda lemma or adjunctions, but I get it now......................
still think it's crazy you can get all this intuition and prove all this stuff about presheaves just from the directed graph/simplicial set case, and I still don't even know any other examples 😭
May 30, 2024 at 3:25 PM
still think it's crazy you can get all this intuition and prove all this stuff about presheaves just from the directed graph/simplicial set case, and I still don't even know any other examples 😭
oh my GOD miraculously against all odds I also managed to prove the co-yoneda lemma today LET'S GOOOOOOOOOO my brain is so fried I need to sleep
May 30, 2024 at 5:31 AM
oh my GOD miraculously against all odds I also managed to prove the co-yoneda lemma today LET'S GOOOOOOOOOO my brain is so fried I need to sleep
finally proved the yoneda lemma..... took Way longer than it should've but I can now die happy
May 29, 2024 at 6:11 PM
finally proved the yoneda lemma..... took Way longer than it should've but I can now die happy
feeling so presheaf-pilled rn
May 29, 2024 at 2:08 PM
feeling so presheaf-pilled rn
just spent my entire day trying to prove this and I'm not even halfway done lmaoooo
so far I've managed to construct the functor C^op x Psh(C) -> Set sending (c,F) to Hom(よ(c),F) so that's some good progress, anyway I'm gonna go to bed
so far I've managed to construct the functor C^op x Psh(C) -> Set sending (c,F) to Hom(よ(c),F) so that's some good progress, anyway I'm gonna go to bed
May 29, 2024 at 4:16 AM
just spent my entire day trying to prove this and I'm not even halfway done lmaoooo
so far I've managed to construct the functor C^op x Psh(C) -> Set sending (c,F) to Hom(よ(c),F) so that's some good progress, anyway I'm gonna go to bed
so far I've managed to construct the functor C^op x Psh(C) -> Set sending (c,F) to Hom(よ(c),F) so that's some good progress, anyway I'm gonna go to bed
yoneda lemma taking up all of my precious drawing time............. worth it tho
May 28, 2024 at 9:55 PM
yoneda lemma taking up all of my precious drawing time............. worth it tho
glad I finally have some good intuition for the yoneda lemma lol, it always bothered me when a text would define what a presheaf is and then immediately jump to yoneda, like why should I care about presheaves what are you even trying to tell me
May 27, 2024 at 4:47 PM
glad I finally have some good intuition for the yoneda lemma lol, it always bothered me when a text would define what a presheaf is and then immediately jump to yoneda, like why should I care about presheaves what are you even trying to tell me
wait omg I'm fathoming so much right now
May 27, 2024 at 4:37 PM
wait omg I'm fathoming so much right now
ok I didn't actually properly think this through the first time I saw it, I thought it would be similar to products of CW complexes, but here you just get the simplicial structure in the product for free???????? what the FUCK
It's really hard to get a sense of imo. I think of them as primarily being there to make products (more generally limits) work out nicely. Think about the product simplicial set Δ^1 × Δ^1, maybe try to work out all of its nondegenerate simplices
May 27, 2024 at 12:54 AM
ok I didn't actually properly think this through the first time I saw it, I thought it would be similar to products of CW complexes, but here you just get the simplicial structure in the product for free???????? what the FUCK
spent literal days trying to prove this which might just be a skill issue on my part LMAO, but in my defense I wanted to prove it in a way that felt natural, and even stating it in a way that didn't feel like a hack required me to construct the isomorphism Fun(J, Fun(C, D)) -> Fun(C, Fun(J, D))
Well limits in presheaf categories are easy! If C is a small category and D is (co)complete then a diagram in Fun(C, D) is a (co)limit iff this is true after evaluating at each object of C. So eg the product of simplicial sets is (X × Y)_n = X_n × Y_n
May 25, 2024 at 8:55 PM
spent literal days trying to prove this which might just be a skill issue on my part LMAO, but in my defense I wanted to prove it in a way that felt natural, and even stating it in a way that didn't feel like a hack required me to construct the isomorphism Fun(J, Fun(C, D)) -> Fun(C, Fun(J, D))