Locally ringed

hangovers feel good when i know it's the last one || ASK ME ABOUT MATH

[messygoku]

bro they send Spain to fucking Chattanooga

is this meme over and over

[sheafified]

Uruguay:

[latinotiktok]

[charl0ttan]

charlotte my tummy hurts how do i fix this :<

SWITCH TO LINUX

[mossquitoman]

I love mathematicians so much. No one else is doing it like them. Group of people genuinely only in it for love of the game. Everyone else hates their subject and thinks its boring and useless and these guys are out here talking about how beautiful and incredible and poetic it is with a twinkle in their eyes. No other subject has such a crazy disconnect between public perception and reality. I love it.

[sheafified]

Loving maths feels like being let in on a secret world we discover and share. You can study things no human can sense no matter how powerful their instruments may be, yet talk to someone and have them recognize the nature of the subject. The greatest story ever told, sprawling out into itself, infinitely vast, carrying on as long as one being out there thinks about it. And most don't want to hear it, and so will never know how it goes past what you know of a popular show you don't watch.

[mossquitoman]

I love mathematicians so much. No one else is doing it like them. Group of people genuinely only in it for love of the game. Everyone else hates their subject and thinks its boring and useless and these guys are out here talking about how beautiful and incredible and poetic it is with a twinkle in their eyes. No other subject has such a crazy disconnect between public perception and reality. I love it.

[sheafified]

Loving maths feels like being let in on a secret world we discover and share. You can study things no human can sense no matter how powerful their instruments may be, yet talk to someone and have them recognize the nature of the subject. The greatest story ever told, sprawling out into itself, infinitely vast, carrying on as long as one being out there thinks about it. And most don't want to hear it, and so will never know how it goes past what you know of a popular show you don't watch.

[what-even-is-thiss]

My computer just randomly stopped running Bluetooth and even digging into the internal files and forcing it to run the software isn’t working for some reason.

[amorphousmassofflesh]

My child just randomly stopped eating, and even hitting them and yelling at them to eat isn't working for some reason.

[what-even-is-thiss]

You know I’ve cycled through like ten different snarky comments I could make in reply to this but frankly this comment (even if it’s some kind of joke) is so out of the ballpark from what I’m talking about that I fear your fundamental understanding of what im referring to isn’t even in the same zip code as the reality of the actual situation and I don’t know how to properly be funny in response to this because I’m so genuinely baffled.

[what-even-is-thiss]

Also I ended up having to upgrade to windows 11 to solve the problem btw

[bloodless-scorpio-33]

okay that's just a tragedy (upgrading to windows 11)

[what-even-is-thiss]

I knew I’d have to do it eventually when they stopped doing security updates for windows 10 but I didn’t think it would be because of my Bluetooth software killing itself in the middle of playing Skyrim with an Xbox controller and then failing to fix the problem after three hours of digging through my computer’s internal files and comments on old reddit posts. That’s not the sort of thing you anticipate taking up your whole day and yet

[sheafified]

I mean, you don't HAVE to use windows 11

[sheafified]

Algebra is for the girls

[gingik]

[fagstaynight]

yeah i dont really like to visit home all too often mostly because of my folks. my mom’s fine but my dad spends all day at spawn giving all the noobs green herb collection quests. he doesn’t even do anything with them he just stuffs them in his pants and comes home with these huge pants and says “well that’s another day in the pocket” like it’s something anyone other than him says. i think my mom only married him because she runs the Lesser Potion Craft Quest down the street that needs green herbs and won’t get anything out of the divorce aside from green herbs. But yeah i’d really appreciate it if you Exterminate 20 Rock Lizard and 3 Elite Rock Lizard

[mathematikerin-freyja]

I hope that when I get into a math phd position I get to take algebraic topology as a course. I want to go in having already familiarised myself with (infinity, 1)-categories and model categories, but no actual concrete examples of either, and then just torment the lecturer with my questions and comments. "Oh shit so this is like a cofibrant replacement in real life! Huh, cute."

[sheafified]

[sheafified]

Thinking about buying an album but shipping is more expensive than the album 😔

[sheafified]

I still don't know what Tor is and at this point i'm too afraid to ask

[sheafified]

I know the definition btw but what are you talking about??

I know what hom and tensor are doing, what are ext and tor doing tho? It's supposed to be like the distance to being inyective / proyective or something like that i think?

[spectracled-eider]

As you're likely already aware, Hom and ⊗ are not fully exact functors (only left and right exact respectively). What Ext and Tor do is tell you what comes next (before resp) in the exact sequence. More explicitly, suppose we have a short exact sequence 0->A->B->C->0, applying Hom(D,-) results in an exact sequence 0->Hom(D,A)->Hom(D,B)->Hom(D,C). But we'd like to know what comes next in the sequence and that's given by the Ext functors. Specifically, the next term here is Ext¹(D,A).

Ext also characterises n-extensions up to a sensible notion of equivalence. For simplicity I'll just explain 1-extensions cause the equivalence relation gets messy otherwise. Given modules A and B, a (1-)extension of A by B is a short exact sequence 0->B->E->A->0. We say two extensions 0->B->E->A->0 and 0->B->E'->A->0 are equivalent if there is a morphism E->E' such that the obvious diagram commutes (I'm currently in bed so I can't draw it, I can if asked). Note that by the Five Lemma, such a map is automatically an isomorphism. Define the set 𝔼(A,B) of extensions of A by B up to the equivalence relation above. It's then possible to make this construction functorial (details are in Hilton and Stammbach) and this functor is naturally isomorphic to Ext¹(-,-) as functors to Set. This means that you can actually put a group operation on 𝔼(A,B) as inherented from Ext¹(A,B). One benefit of this perspective is Ext can be defined on Abelian categories that don't even have enough projectives/injectives.

To my knowledge, Tor doesn't have a similar sort of construction.

There is also a homotopy theoretic interpretation of what Ext and Tor are doing! I'll avoid taking about model categories in general but this is the more general setting in case anyone was interested. The derived category of an Abelian category A, e.g. of R-Mod, is the localisation of the category of cochain complexes of A, Kom(A), with respect to cohomology isomorphisms. That is, we consider the category whose objects are still cochain complexes but where cohomology isomorphisms are genuine isomorphisms. More formally, we have a functor Q:Kom(A)->D(A) which satisfies the following universal property: any functor F:Kom(A)->C, where C can be any category, which sends cohomology isomorphisms in Kom(A) to isomorphisms in C can be factored through Q. Of course this defines D(A) up to isomorphism though it doesn't necessarily guarantee existence. However we can construct D(A) another way using model category theory (specifically it's the homotopy category of Kom(A) w.r.t a specific model structure). Now note that we can embed A into Kom(A) by taking mapping an object of A to the cochain complex which is that object in degree 0 and 0 everywhere else. So applying Q, we actually embed A into D(A). So one might ask whether we can extend functors defined on A to functors defined in D(A). For example, if A=R-Mod, Hom and ⊗ are functors to Ab, and we can ask whether they can be extended to a functor D(R-Mod)->D(Ab). Then answer is yes: we take derived functors! That is, there is a specific construction one can do using model category theory to extend functors to the homotopy categories (this is called the total derived functor). When we apply that construction here, we get a cochain complex (an object in D(Ab), and we can take the cohomology of this complex and we retrieve the homological algebraic derived functors! (Note: I think depending on whether the functor is right or left exact and co/contravariant you might need to do this on chain complexes rather than cochain complexes). So the homotopy theoretic answer is that Ext and Tor serve to extend Hom and ⊗ to the relevant derived/homotopy categories!

[sheafified]

I still don't know what Tor is and at this point i'm too afraid to ask

[sheafified]

I know the definition btw but what are you talking about??

I know what hom and tensor are doing, what are ext and tor doing tho? It's supposed to be like the distance to being inyective / proyective or something like that i think?

[sheafified]

I still don't know what Tor is and at this point i'm too afraid to ask

[tpwrtrmnky]

There's cultural relativism and then there's naive anglocentric cultural dipshitism where you go "well, they have a word for a thing you put on your foot to protect it from the environment, but that word doesn't NECESSARILY translate to shoe because they make them differently from how we make shoes, so really they don't have a concept of a shoe the way we understand shoes, and also they have a more spiritual understanding of covering your feet when going outside, so really it's completely different" and then you ask someone from the culture in question, and they're like "yeah that word just means shoe."

[sheafified]

Una vuelta vi un infographic que decía que las culturas latinoamericanas son relativistas con respecto al tiempo porque cuando se dice que algo es a las 15 se espera que arranque ~15:30. Terminaba diciendo "a la gente de culturas relativistas no le importa que algo se haga a tiempo, si no que se haga bien"

Todo este continente está atado con alambre, y llegamos tarde porque si llegas en hora te ponen a fregar

[galois-groupie]

I'm not a category theorist, I'm a category engineer

[mathematikerin-freyja]

EXPLAIN? EXPLAIN! EXPLAIN! EXPLAINNNN!

[galois-groupie]

It's like the computer scientist/engineer distinction. I don't study the essential nature of categories, I take some interesting question, define a random category, and then straighten/unstraighten until I get the answer which I want.

[mathematikerin-freyja]

I would be interested to see this in action. Can you point me in the direction of a representative paper demonstrating this style of working with categories

[sheafified]

BMRRT or BMR 2006, representation theory:

I want to study the modules of a given algebra A, i construct the "cluster category C" A and prove the endomorphisms of certain objects in C will yield (almost) the same modules as the original algebra.

[messygoku]

u ever feel like the aguará Guazú que encontraron en medio de Entre Ríos

just out of place as fuck

[sheafified]

Los amo pero si un día me cruzo uno en la calle me da algo