#fixed point
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Welcome to the premier of One-Picture-Proof!
This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)
Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)
The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.
Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)
Preliminaries and Definitions:
B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).
The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)
φ is just some morphism A→B^A.
Δ is the diagonal, which maps a↦(a,a).
1 is the terminal object, you can think of it as a single-point set.
We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:
Exercises:
What is the statement of the theorem?
Work through the proof, follow the arrows in the diagram, understand how it is composed.
What is the more popular name for this technique?
What are some applications of it? Work through those corollaries in the diagram.
Can the theorem be modified for epimorphisms? Why or why not?
For the advanced: What is the precise requirement on the category, such that we can perform this proof?
For the advanced: Can you alter the proof to lessen this requirement?
Bonus question: Can you see the Sicko face? Can you unsee it now?
Expand to see the solutions:
Solutions:
This is Lawvere's Fixed-Point Theorem. It states that, if there is a point-surjective morphism φ:A→B^A, then every endomorphism on B has a fixed point.
Good job! Nothing else to say here.
This is most commonly known as diagonalization, though many corollaries carry their own name. Usually it is stated in its contraposition: Given a fixed-point-less endomorphism on B there is no surjective morphism A→B^A.
Most famous is certainly Cantor's Diagonalization, which introduced the technique and founded the field of set theory. For this we work in the category of sets where morphisms are functions. Let A=ℕ and B=2={0,1}. Now the function 2→2, 0↦1, 1↦0 witnesses that there can not be a surjection ℕ→2^ℕ, and thus there is more than one infinite cardinal. Similarly it is also the prototypiacal proof of incompletness arguments, such as Gödels Incompleteness Theorem when applied to a Gödel-numbering, the Halting Problem when we enumerate all programs (more generally Rice's Theorem), Russells Paradox, the Liar Paradox and Tarski's Non-Defineability of Truth when we enumerate definable formulas or Curry's Paradox which shows lambda calculus is incompatible with the implication symbol (minimal logic) as well as many many more. As in the proof for Curry's Paradox it can be used to construct a fixed-point combinator. It also is the basis for forcing but this will be discussed in detail at a later date.
If we were to replace point-surjective with epimorphism the theorem would no longer hold for general categories. (Of course in Set the epimorphisms are exactly the surjective functions.) The standard counterexample is somewhat technical and uses an epimorphism ℕ→S^ℕ in the category of compactly generated Hausdorff spaces. This either made it very obvious to you or not at all. Either way, don't linger on this for too long. (Maybe in future installments we will talk about Polish spaces, then you may want to look at this again.) If you really want to you can read more in the nLab page mentioned below.
This proof requires our category to be cartesian closed. This means that it has all finite products and gives us some "meta knowledge", called closed monoidal structure, to work with exponentials.
Yanofsky's theorem is a slight generalization. It combines our proof steps where we use the closed monoidal structure such that we only use finite products by pre-evaluating everything. But this in turn requires us to introduce a corresponding technicallity to the statement of the theorem which makes working with it much more cumbersome. So it is worth keeping in the back of your mind that it exists, but usually you want to be working with Lawvere's version.
Yes you can. No, you will never be able to look at this diagram the same way again.
We see that Lawvere's Theorem forms the foundation of foundational mathematics and logic, appears everywhere and is (imo) its most important theorem. Hence why I thought it a good pick to kick of this series.
If you want to read more, the nLab page expands on some of the only tangentially mentioned topics, but in my opinion this suprisingly beginner friendly paper by Yanofsky is the best way to read about the topic.
#mathblr#mathematics#set theory#diagram#topos theory#diagonalization#topology#incompleteness#logic#nLab#Lawvere#fixed point#theorem#teaching#paradox#halting problem#math#phdblr#Yanofsky#Cantor#Tarski#Gödel#Russell#philosophy#category theory
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Tbh there are many gfl moments I wish had cgs or moments I wish were shown differently, like the only thing I can pitch in for "shown differently" is a front view of angelia in the actual moment of ct confrontation and not before cause it would go so hard, but stuff like J discovering the blood splatter at the mansion, a shadowed side view of bram fucking up with Mona, Morri casually destroying narcis, the surgery scene, an actual scene from the fp prayer scene would have been so fucking great
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some dunmeshi restaurant au doodles
#dungeon meshi#laios touden#chilchuck tims#kabru dungeon meshi#dunmeshi#illustration#artists on tumblr#fanart#reupload cause my deuteranomaly rlly popped out in the last version#whoever pointed out that i made them green accidentally i owe u my life#its fixed now. i think. i cant tell
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many centuries down the line they can manage to have a civil enough family game night.
#cult of the lamb#narilamb#Leshy carved losers corner above lambs bed many games ago and they think it’s too funny to fix#Anyway lamb does eventually marry narinder but after a good while of tormenting him on who to choose for marriage#I think at one point narinder starts dragging suitors into the battle pit#And says only someone who can beat him is good enough for lamb yayyy#This comic came out pretty clumsy but I’m just glad it’s done#cotl art
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going through my old journals as part of therapy homework and i'm reading a section written in the emotional wreckage of a full-on breakdown when i get hit with this line:
There is never a satisfying answer to ‘Why didn’t they love me?’
like wow babe. good fucking point
#like you were on the ground biting the carpet and dry sobbing while you wrote that and still. good fucking point#not a shitpost#cptsd#and it's true. there's never a satisfying answer#the truth is i know why i wasn't loved#i analyzed my parent's traumas and abuse to death. i understand why i alienated and was alienated from my siblings#i know why my mom was too overwhelmed to be capable of nurturing#i know why my dad vanished into addiction and avoidance#the details of our cycles of trauma and cptsd and family history i have a phd in all of it#i understood perfectly. i spent years studying and now i knew the answer#and guess what? IT WAS NOT SATISFYING!!!#because they still didn't love me! and i still couldn't change that!#it was still a completely unsatisfying state of affairs!#so like. when the people who are supposed to love you...don't.#when the people who are supposed to take care of you...fail to#you can look for answers and reasons and explanations#but that's not actually going to FIX your situation.#and it's probably not within your ability TO fix the situation. (and definitely not your job)#because you don't need answers--you need a new situation#*inserts Just Walk Out. You Can Leave!!! (Running Skeleton) Meme*#and yes. walking out isn't always possible.#but for you i hope it will be one day soon. and i hope you build the courage to take that leap.#stepping away from the people who failed to love you...it feels like being untethered but also like being lighter than air#new and scary. immensely relieving. the future opens up. empty but empty like a canvas. blindingly bright until your eyes adjust#like climbing out of a pit you called home and for the first time realizing how bright the light of day can truly be#when you aren't just getting glimpses from the bottom of a hole
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#selfie bee#good evening friends!! how are you doing! C:#I'm very very sleepy I got a new ikea office chair and I build it all myself#I think it went okay! I don't think I pulled the back screw tight enough and now the back is a bit loose#I can probably fix it but I can also ignore it for the next 18 years#thats how long the old chair held up!! in germany it could now drink vodka and drive a car!!#not at the same time that is illegal! not at the same time!! (❁´▽`❁)*✲ﾟ*#but the day is not over yet my uncle asked me for a big art quest and I do not want to disappoint#he wants a muppet tattoo and asked me to draw it#my uncle has started to get tattoos a few months ago#as far as I know he has now gotten 3 note clefs 3 stars a flower and multiple birds#he also started getting piercings but so far I managed not to know exactly where#I think tattoos are super cool (´｡･v･｡｀) I wish I had a good idea for a tattoo but the last time I was very sure about getting a tattoo#it was heath ledgers face as the joker#at that point I was 12 and would not see the actual movie for two more years#a muppet tattoo is a way better idea!! he asked for the count van count! that is also one of my top 3 muppets ₍՞◌′ᵕ‵ू◌₎♡#I always thought I knew a lot about muppet lore but since I started looking up muppet pictures I think there are still a lot of secrets#can the muppets from the Sesame Street actually leave the Sesame Street?#I think Kermit is both on the Muppet Show and on Sesame Street but he is also like the boss muppet#he might have special abilities#I hope you're having a good day friends!! C:#I think I'll post a Sherlock comic later this week#miss you!! ♥♥♥
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there's something so intensely lovely about the TARDIS whoosh whoosh being the same. you're watching doctor who in 1971? it whoosh whoosh. jump forward to 2005 and guess what? it's the same whoosh whoosh. the doctor's face may change, the budget may grow, the theme may be updated, but the whoosh whoosh of the tardis remains
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last three seasons spn are crazy. its just dean being borderline suicidal while sam tries to fix it by basically dangling his keys at him and going : “dean look! cowboys!” “dean look! strip club!” “dean look! haunted action figure.”
Meanwhile castiel is like i see that dean is suicidal, this is clearly my fault so i will remedy this by dying.
#and then dean is fixed by jack getting him a yogurt from the fridge or some shit#dean: yeah i just dont believe theres any point living any more. sam: huh dean are you ok? that sure was a funny thing to say#dean: yeah i just dont believe theres a point to life anymore. cas: don’t worry dean. i’ll always be here. *dies*#dean:yeah i just don’t believe theres a point to living anymore. jack: bummer. want to play mario kart?#dean: sure kid why not#destiel#dean winchester#spn#sam winchester#supernatural#but seriously#i feel like we dont talk about how dean was trying to die at the slightest inconvenience in the later seasons and its like??? bestie???
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Had this idea way back when we first saw the Grimwalker graveyard in King’s Tide. The chances of this happening a pretty slim, but i wanted to get the idea out anyway
Had so much fun with the composition and speech bubbles of this, i gotta gotta gotta draw some more Collector art at some point
#this was a lot of fun but dear lord this was genuinely one of the hardest things ive ever had to colour#the owl house#TOH#TOH fanart#toh spoilers#the owl house spoilers#toh hunter#toh the collector#comics tag#not gonna post the rest of the hexsquad but you know ...... theyre there#Belleskiart#EDIT: fixed a spelling mistake#thankyou sm person who pointed that out
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in paris you gotta sleep with one eye open
#she smoked too tough. her swag was too different. her bitch was too bad. they’re gonna kill her#kaitsart#gif#interview with the vampire#interview with the vampire 2022#interview with the vampire fanart#iwtv#claudia de pointe du lac#claudia iwtv#delainey hayles#fun fact i actually finished this yesterday but didnt wanna double post lol#no one point out the eensy weensy mistake i’m not fixing it 😔😔
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🦋
#dimension 20#mentopolis#conrad schintz#the fix#justin fication#just one more little sketch#because today’s the rare day i can draw#and they’re hitting every single point that i love and will bawl over#fanart#my art
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amy and rory chilling at home in between trips, summer of 2012, watching the olympics. stadium’s suddenly empty and they’re both waiting for the sound of the tardis in the backyard any second now. oop, wait, crisis averted apparently, guess the doctor didn’t need to intervene after all lol, wasn’t that odd? oh dunk, some rando just grabbed the torch off the fallen torchbearer! ah well, this might as well happen, the opening ceremony is already so goddamn weird. haha that guy’s a little over dressed for a run-
wait
those pants. rory i know those pants. rory
#if they were separated during that point shh don’t worry about it#all i’m saying is the girl who waited and made countless drawings and dolls of the raggedy man will never forget his pinstriped pants#i’m fairly certain they knew the doctor could regenerate and maybe even seen his past faces#but they wouldn’t remember that nearly as well as amy would remember those exact pants#and the combination alien activity + no visit from Their doctor to go fix it#it’s not a huge jump to land on that conclusion#but i don’t think rory would be as fully convinced as amy#doctor who#i know this exact post was probably made 13 years ago but that’s never stopped me before#amy pond#rory williams
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Red White and Royal Blue movie + book moments
#rwrb movie#red white and royal blue movie#alex claremont diaz#henry fox mountchristen windsor#zahra bankson#nicholas galitzine#taylor zakhar perez#firstprince#rwrb#red white and royal blue#gifs#mine#gifs: alex#gifs: henry#gif: zahra#pls ignore the typos#I will fix it at some point#I was overly excited about this gifs
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Happy Valentine's Day from the Young Justice group! 💗
#young justice 1998#young justice#tim drake#kon el#bart allen#greta hayes#cassie sandsmark#cissie king jones#dc comics#my art#digital art#these took me way too long and at some point I lost all motivation#could go back in and fix a million things but it's good enough!
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tiny cuddling skk for the soul
Dazai's clingy/tactile nature set free, and Chuuya who's still learning how to accept and reciprocate
#the balancing continues (though i think it's almost 50/50 at this point i surprised myself by scrolling through my art)#all in the same post for maximum concentration of skk cuddles (also to let me fix some things that were bothering me)#and of course dazai still cheats this changes nothing#bsd#bungo stray dogs#bungou stray dogs#bsd fanart#bsd chuuya#bsd nakahara chuuya#bsd dazai#bsd dazai osamu#skk#soukoku#nawy's doodles
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putting my prediction on record now that the coming decade is going to see the rise of viral-marketed fancy at-home water filtration systems, driving and driven by a drastic reduction in the quality of U.S. tap water (given that we are in a 'replacement era' where our current infrastructure is reaching the end of its lifespan--but isn't being replaced). also guessing that by the 2030s access to drinkable tap water will be a mainstream class issue, with low-income & unstably housed people increasingly forced to rely on expensive bottled water when they can't afford the up-front cost of at-home filtration--and with this being portrayed in media as a "moral failing" and short-sighted "choice," rather than a basic failure of our political & economic systems. really hope i'm just being alarmist, but plenty of this already happens in other countries, and the U.S. is in a state of decline, so. here's praying this post ages into irrelevance. timestamped April 2023
#apollo don't fucking touch this one#serious post#not a shitpost#hope i forget about this post and have no reason to ever look back on it one day#fyi i'm aware that access to potable water is already a major issue in parts of the U.S. yes i know flint michigan exists#i'm saying that this issue is going to GROW unless local & federal governments work together to fix it.#so it's a matter of if we trust them to fix it. And well--do you?#what are the chances the government just denies there's a problem until the water actually turns brown#at which point it's already been common knowledge for years and people have just become resigned and that's our new normal#i'm mean come on. how many of us already believe that we're being exposed to dangerous pollutants we don't know about and can't avoid#like that's pretty much just part of being a modern consumer. accepting that companies will happily endanger your life for a few pennies#and the most you'll get is like a $50 gift card as part of a class action rebate 20 years down the line#probably the history books will look back on Flint as a warning and a harbinger that went ignored#luxury condos will advertise their built-in top-of-the-line filtration systems--live here and you can drink water straight from your tap!#watch the elite professional class putting $700 dyson water filtration systems on their wedding registry#while the rest of us figure out how to fit water delivery into our grocery budget while putting 90% of our paycheck towards rent#also eggs are $15
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