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bubbloquacious · 7 months

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Hydrogen bomb vs. coughing baby: graphs and the Yoneda embedding

So we all love applying heavy duty theorems to prove easy results, right? One that caught my attention recently is a cute abstract way of defining graphs (specifically, directed multigraphs a.k.a. quivers). A graph G consists of the following data: a set G(V) of vertices, a set G(A) of arrows, and two functions G(s),G(t): G(A) -> G(V) which pick out the source and target vertex of an arrow. The notation I've used here is purposefully suggestive: the data of a graph is exactly the same as the data of a functor to the category of sets (call it Set) from the category that has two objects, and two parallel morphisms from one object to the other. We can represent this category diagrammatically as ∗⇉∗, but I am just going to call it Q.

The first object of Q we will call V, and the other we will call A. There will be two non-identity morphisms in Q, which we call s,t: V -> A. Note that s and t go from V to A, whereas G(s) and G(t) go from G(A) to G(V). We will define a graph to be a contravariant functor from Q to Set. We can encode this as a standard, covariant functor of type Q^op -> Set, where Q^op is the opposite category of Q. The reason to do this is that a graph is now exactly a presheaf on Q. Note that Q is isomorphic to its opposite category, so this change of perspective leaves the idea of a graph the same.

On a given small category C, the collection of all presheaves (which is in fact a proper class) has a natural structure as a category; the morphisms between two presheaves are the natural transformations between them. We call this category C^hat. In the case of C = Q, we can write down the data of such a natural transformations pretty easily. For two graphs G₁, G₂ in Q^hat, a morphism φ between them consists of a function φ_V: G₁(V) -> G₂(V) and a function φ_A: G₁(A) -> G₂(A). These transformations need to be natural, so because Q has two non-identity morphisms we require that two specific naturality squares commute. This gives us the equations G₂(s) ∘ φ_A = φ_V ∘ G₁(s) and G₂(t) ∘ φ_A = φ_V ∘ G₁(t). In other words, if you have an arrow in G₁ and φ_A maps it onto an arrow in G₂ and then you take the source/target of that arrow, it's the same as first taking the source/target in G₁ and then having φ_V map that onto a vertex of G₂. More explicitly, if v and v' are vertices in G₁(V) and a is an arrow from v to v', then φ_A(a) is an arrow from φ_V(v) to φ_V(v'). This is exactly what we want a graph homomorphism to be.

So Q^hat is the category of graphs and graph homomorphisms. This is where the Yoneda lemma enters the stage. If C is any (locally small) category, then an object C of C defines a presheaf on C in the following way. This functor (call it h_C for now) maps an object X of C onto the set of morphisms Hom(X,C) and a morphism f: X -> Y onto the function Hom(Y,C) -> Hom(X,C) given by precomposition with f. That is, for g ∈ Hom(Y,C) we have that the function h_C(f) maps g onto g ∘ f. This is indeed a contravariant functor from C to Set. Any presheaf that's naturally isomorphic to such a presheaf is called representable, and C is one of its representing objects.

So, if C is small, we have a function that maps objects of C onto objects of C^hat. Can we turn this into a functor C -> C^hat? This is pretty easy actually. For a given morphism f: C -> C' we need to find a natural transformation h_C -> h_C'. I.e., for every object X we need a set function ψ_X: Hom(X,C) -> Hom(X,C') (this is the X-component of the natural transformation) such that, again, various naturality squares commute. I won't beat around the bush too much and just say that this map is given by postcomposition with f. You can do the rest of the verification yourself.

For any small category C we have constructed a (covariant) functor C -> C^hat. A consequence of the Yoneda lemma is that this functor is full and faithful (so we can interpret C as a full subcategory of C^hat). Call it the Yoneda embedding, and denote it よ (the hiragana for 'yo'). Another fact, which Wikipedia calls the density theorem, is that any presheaf on C is, in a canonical way, a colimit (which you can think of as an abstract version of 'quotient of a disjoint union') of representable presheaves. Now we have enough theory to have it tell us something about graphs that we already knew.

Our small category Q has two objects: V and A. They give us two presheaves on Q, a.k.a. graphs, namely よ(V) and よ(A). What are these graphs? Let's calculate. The functor よ(V) maps the object V onto the one point set Hom(V,V) (which contains only id_V) and it maps A onto the empty set Hom(A,V). This already tells us (without calculating the action of よ(V) on s and t) that the graph よ(V) is the graph that consists of a single vertex and no arrows. The functor よ(A) maps V onto the two point set Hom(V,A) and A onto the one point set Hom(A,A). Two vertices (s and t), one arrow (id_A). What does よ(A) do with the Q-morphisms s and t? It should map them onto the functions Hom(A,A) -> Hom(V,A) that map a morphism f onto f ∘ s and f ∘ t, respectively. Because Hom(A,A) contains only id_A, these are the functions that map it onto s and t in Hom(V,A), respectively. So the one arrow in よ(A)(A) has s in よ(A)(V) as its source and t as its target. We conclude that よ(A) is the graph with two vertices and one arrow from one to the other.

We have found the representable presheaves on Q. By the density theorem, any graph is a colimit of よ(V) and よ(A) in a canonical way. Put another way: any graph consists of vertices and arrows between them. I'm sure you'll agree that this was worth the effort.

#math #adventures in cat theory #oh btw bc よ is full and faithful there are exactly two graph homomorphisms よ(V) -> よ(A)#namely よ(s) and よ(t)#which pick out exactly the source and target vertex in よ(A)

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ethies · 7 months

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bwww ... does the knight of space mind coming up w lucky cat themed names , pronouns && titles ? ^^ ( if not too much or you do not feel comfey doing the lucky cat npts , pronouns that r similar 2 she / her or he / him would b nice too :3 )

hello ! here are some names , pronouns , and titles for you ...! this divine being thanks you for requesting , schy had lots of fun writing these out ^^

𓇼 . . lucky cat names : luckitty , fortune , fausta , faustette , faustina , felicity , kismet , kizzy , kit , katte , whisker , calico , callie , cheshette , cheshire , mewlina , meowli , mitten ( s ) , lioness , lionette , sylvie , sylvester , purrlina , purrette , nepeta

𓇼 . . lucky cat pronouns : mew ﾉ mews , purr ﾉ purrs , mrr ﾉ mrrp , mao ﾉ maos , meo ﾉ meow , whis ﾉ whisker , lu ﾉ luck , for ﾉ fortune , fa ﾉ fate , paw ﾉ paws , fu ﾉ fur , paw ﾉ claws , mrow ﾉ mrows , leap ﾉ leaps , pitter ﾉ patter , cat ﾉ nip , hi ﾉ hiss , ble ﾉ bless , chan ﾉ chance , des ﾉ destiny , :3 ﾉ :3s

𓇼 . . lucky cat titles : che who meows / mews / purrs / etc , this lucky feline , this fortunate kitty , the one who prospers , this cat of fate , the one you have met by chance , che who ' s luck flows like a river , cher luck , cher kismet , che of unmeasured fortune , this cat who walks with luck on cher side , the feline who is made of luck , one who is the most fortunate , che with strong destiny

𓇼 . . she he pronouns : shy ﾉ hyr , schy ﾉ schyr , sh<3 ﾉ h<3r , shx ﾉ hxr , sae ﾉ saer , shwe ﾉ hwer , sh! ﾉ h!r , sh? ﾉ h?r , sh* ﾉ h*r , sh♡ ﾉ h♡r , se ﾉ shim , hy ﾉ hym , hy ﾉ hymn , hey ﾉ hem , h<3 ﾉ h<3m , hx ﾉ hxm , hae ﾉ haem , hwe ﾉ hwim , h! ﾉ h!m , h? ﾉ h?m , h* ﾉ h*m , h♡ ﾉ h♡m , he ﾉ her , she ﾉ him , e ﾉ im , e ﾉ er , her ﾉ sim

please Enjoy these ...! the One Who embodies purity Worked very hard On Them ... v? is especially Proud of the titles ! ae Hopes they are all to your satisfaction , and this Sea creature thanks You for Requesting ! ^^ che may be Replaced with Anybeings Individual pronouns , as always~

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achillean-knight · 7 months

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LISTEN

This was originally going to be a shit post bc I saw a comic panel (idfk what comic it was from or if it was just fan made) and I had an urge to draw Noir in a 1930's Evening Gown for shits and giggles and totally not project but then it turned into a serious study of this one dress I found and I was like

"Damn. I'm spending wayyy too much time on this goddamn"

#spider noir #Don't come at me for this idfk his personality besides what ive seen in the movie and the fact he named his cat Ding Ding #IDK if he'd even wear dresses #it's the 1930's i don't even think it's legal #But the image has been rotting my brain and I had to.#Yes I added Ding Ding bc I can not get over her(?)#spiderman noir #ooc?#ooc.#1930's #Noir #Evening gown #YIPPEEE #Shitpost?#Don't kill me for this aight #my art

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diathadevil · 8 months

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HAPPY BIRTHDAY MY GIRL 😭😭😭 I had this whole idea where Ayushiki got a cat later in life and she's a spoiled rotten baby.

#corpse party #ayumi shinozaki #yoshiki kishinuma #ayushiki #fan art #digital art #art #illustration #yes he's made a giant custard caramel pudding for her yes it's absolutely a giant caramel pudding and im so jealous I made it #because i want to EAT IT TOO AUGH #Anyway the cat's name is Her Majesty and she;s a black norwegian forest cat #EDIT: I FORGOT AYUMI'S HAIR BITS STICKING OUT on the first drawing so I updated it 😭😭😭 im a professional

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djpurple3 · 3 months

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Something about two teens who've devoted their everything to trying to preserve what's left of a destroyed world discovering what they are actually protecting...

Or: two besties gawking at the first butterfly they've ever seen

#silver the hedgehog #blaze the cat #sonic the hedgehog fanart #silver the hedgehog fanart #blaze the cat fanart #sonic 06 #i dont know if its correct ettiquite to tag sonic fanart under sonic's name bc its the franchise or not ^-^'#their poses r a bit wonky but i still think theyre cute #djpurple3's art yo #traditional art

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lucalicatteart · 11 months

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Finally finished my weird hanging painting thing (originally a secondhand partially-done 'paint by numbers' kit that I found at a thrift store and kept to repurpose lol)! Imagery somewhat based in my own worldbuilding projects, and text written in my constructed language for one of my fantasy species, but also vaguely inspired by old tapestries and illuminated manuscripts and etc. I've never been great at neat clean patterning or text, but it looks cool from afar, and I always enjoy making "props" or things that are somewhat like real objects that might could exist in my world. :0

(additional pictures/info under the readmore)

Here's what it was originally! I probably didn't have to actually have a river running down the middle because it further makes the composition of the whole thing weird (various connected yet separate locations and things happening, instead of one unified event being portrayed), but I wasn't sure if I'd be able to fully cover up the already existing paint that was there.. and I can also kind of justify it by going with a more "all the imagery is just symbolic so it doesn't have to make exact sense" approach lol.. How is one half of the grass green and the other is suddenly snowy? shhhh.. it's not literal.. shhh...

Made a vague sketch, then painted over it, and then added more distinct lines in black pen. Center image first and border second.

The very last thing was the text, which actually took forever to translate because my conlang is still only like.. partially done, and some of the grammar is not worked out exactly how I would like it to be, so a few sentences I had to think about for a long time before just going "eh, this is probably not how I would do it if I considered it more, but I'll go with it for now" lol . I also am not entirely satisfied with all of the characters for the writing system, but again, it's good enough for a quick project, it doesn't have to be 100% accurate and perfect because it's a fake language that nobody knows anyway lol.

I thought about breaking down the text and translation here like I have for some of the tidbits of Avirrekava (the language) in things I've posted in the past, but I think it would take too long and is not interesting to anyone but me ghghj, so for the sake of getting the post out quickly, I shall not spend an hour typing All That lol.

The general jist of the writing though is that it's just about the Avirre'thel being cast out from the other elves, after abandoning their magic for immortality as a means to truly attain perfection (an important concept in elven culture), the usual, blah blah blah, but how it's Actually A Good Thing, because the gods are wrong and immortality is Cool actually and they like the shitty frozen lands they were sent to, so it's fine that everyone else is being a Hater about it lol

Lastly, here's a few photos outside in the sun to TRY and show the gold detailing actually shimmering or showing up! It really doesn't come through in photos, but there's plenty of little golden spots to highlight light or Importance.

Mostly the fire, the pink sparkle that represents magic, the red drop that represents blood, the light behind Inaashi's hands and head (common symbol for the elven religion/one of their main gods, shout out to anyone who read the ancient elven religion post and recognized that lol), the sun, and the symbol for the Avirre'thel/country of Navyete at the very top. I did a few other gold bits, but they're not highlighted because they're Significant, more just that it looked more symmetrical to have some gold on the border too lol.

Other things of note: The animals are not actually significant to Avirre'thel culture really, I just wanted to put a cat and a bird because I like them lol. (I also wanted to have a few funny looking creatures, as I was slightly trying to go with the 'in some old medieval painting the anatomy and perspective is very weird' vibe, though I think some of the other parts of it look too Normal to pull it off entirely). Same with the four leaf clover, which means nothing in their culture - but these are the only areas where stuff was just added self-indulgently .

Bligabata (giant cabbage that grows along rivers in Navyete) making an appearance! The architecture of the building IS based on actual concepts for ancient elven/older Avirre'thel architecture and metalwork. The Avirre'thel who's turning away from Inaashi/elves/magic and collecting blood, is doing so in a Special Bowl, as is part of their culture (collecting it in the hands, or just in a normal vessel would be disrespectful, they have Specific Bowls which is the only thing blood can be kept in, etc.).

The figure that represents Jhevona (and thus, a closer connection to magic, celestial imagery, etc.) is in weird ugly teal, which is not necessarily a color or design associated with them, as I don't have much common culture (like clothing) worked out for Northern Jhevona (who the avirre'thel would have come into contact with) yet, BUT everyone else is in more Typical colors (a northern elf in green, Inaashi in lavender + white + blue, an Avirre'thel in darker purples and reds).

Some things, like the four figures in the corners, and the two people + fish in the stream, do not currently have a meaning, but in-world they would.. Like, I could make up lore for how they're culturally significant and it would be true because I am god of the world, but I don't have anything currently. But just know.. they DO mean something, I just haven't decided it yet, maybe kind of fill in as I go, come up with a meaning later lol. Probably along the lines of an old myth from the ancient elven religion, a story, etc.

I don't know, probably other stuff, but that's my Trying To Keep It Short rambling for now lol. I'm just glad I finally finished this! For how vaguely sloppy it is up close (everything being completely freehanded, only used rulers once when doing the initial sketch and lining where the border should be + my hands are shaky + the canvas is bumpy + my handwriting is scratchy and terrible + etc. etc.) it still took a REALLY long time, even when not trying to make it all perfect. Especially if including the text translation + writing, which took like 3+ hours itself.

Maybe all the asymmetry/lack of things being centered is NOT because I was too lazy to measure anything, but is actually because in-universe, it's a practice illustration made by some young apprentice who has to work on little canvases for years before he can be trusted will a full sized mural or tapestry. It's his first week on the job! of course he's uncoordinated! don't laugh at him!!! lol

#worldbuilding #elves #I AM WORKING ON A NEW PAVENTURE POST also !!!! I know I keep being like 'oh I'm going to get back to that! I'll stick to it this time!!'#and then another whole month goes by without me posting a new poll adventure - however - this time I DID fully intend to so #*do another one soon but my beloved beautiful perfect cat unfortunately passed away AND there was a heat #wave ANd I felt sick for a while for unrelated reasons so I just genuinely was not focused on posting online at all #I am trying to get back to it though along with other things hopefully so.#ANYWAY #avirre'thel #irithoas #maybe???? not super relevant to elves but I'll keep it intheir tag anyway also. Just since their lore is so closely tied with avirre'thel s #stuff and they're mentioned in the post. Or the gods are. Inaashi is.#OIGUGUGUGUHH I should have done a tapestry with the FCJhjkING triplets!!!!! Sehalanora Semoniyare and the other one whatever the hell #his name is. ... sehalanora my beloved .. (I'm referencing the ancient elven gods - for those who dont know)#It's funny that I rarely watch tv shows and when I do I rarely if EVER care about characters at all in any capacity #with maybe like a handful of even then extremely minor exceptions so I cannot relate to the concept of like 'having a blorbo' or whatever #but then for my extremely niche worldbuilding content #.. it's like OMG MY FAVORITE character!!! my favorite obscure god from a religion #that I entirely made up myself for a cultural group that I also made up that literally only I and maybe like two other #people who are able to sit through my novel long dry and wordy worldbuilding posts care about!! you all know them DUH!!#even WITHIN modern elven culture in the world at the moment in current day most people do not give a shit about them hghj #BUT .. I should have made a painting of the siblings actually!!! I stand by that!!#I mean I like Inaashi and Nisateyu and everything too. Actually all of them are fine except for Ea'ivuyera I guess. whoever the #like War and Order bootlicker god is basically. and the Evil dumbass one. but all the others are fine. I'm suprised I'm even able to rememb #that many ancient elven goofily long names ghgh.. But I could have maybe made it about the elven gods #The thing is just that.. i Don't have ancient elvish worked out as a language and I knew I wanted to put text on it #so it kind of HAD to be something written/drawn by the Avirre'thel #Knwoledge of the ancient elven gods is still a thing in their culture. But usually more as a joke or just a common fairytale knowledge #sort of thing. not really something to make a painting of. Inaashi is here less because of Inaashi The God being genuinely significant and #and more just she's there to Symbolize the elven religion as a whole. just like all the other figures are mere symbols of things. etc.

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junkyard-gifs · 8 months

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Else how can he keep up his tail perpendicular?

Or spread out his whiskers? or cherish his pride?

The boys doing some synchronised tail-twirling and preening: Bennett Ryan as Tumblebrutus, Jack Duff as Skimbleshanks, and Jarrad Green as Tugger. Oasis cast 13, May 2023 (X).

#s: naming of cats #tumblebrutus #skimbleshanks #rum tum tugger #jarrad green #jack duff #bennett ryan #c: tumblebrutus #c: skimbleshanks #c: rum tum tugger #skimbleshanksxtugger #skimbleshanksxtumblebrutus #230500 #cats the musical #oasis #oasis 13

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warriors-ideas · 5 months

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au where snowkit lives to be the prophecy cat instead if brambleclaw

Sadly i Don't think the erins themselves would write this in canon properly. So i will write it myself

🌨️

#warriors #warrior cats #snowkit #brambleclaw #wc #warriors au #warrior cats au #wc au #au ideas #other’s ideas #ideas #i think i remember seeing an au like this a while back (brambleclaw lived but was named brambleflower and was snowclaw [I think that was sno #wkit’s warrior name]’s mate)#it used to be one of the my fav aus a while back but i can’t remember who made it or the post #also sorry for the big gaps between posts everyone #just busy things happening so i haven’t been posting as much

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wisecrackingeric-2 · 1 year

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Everyone calling Leon S Kennedy a catboy gave me a vision

Leon and Ashley as warrior cats

Bonus:

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space-writes · 24 days

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find the word

tagged by @jmhwritesstuff, thank you! my words are room, point, stars, and blue. A Question of Trust is coming along nicely enough to find all of these now!

no-pressure tagging @memento-morri-writes @serenanymph @zmwrites and @sam-glade with the words take, lower, fine and dive.

room

Her Sir smoothed her hair back over her shoulders, and kissed the tip of her ear. “I will return when the room is ready, salen’cath.”

point

“Pulling my hair is a form of kiss now, is it?” “You liked it.” He gasped as Rizeth hauled him closer, their lips an inch apart. “You did, I heard you.” “Whether or not I liked it is beside the point, you—” he cut himself off, and Ashenivir wanted to bite him, and not just to be a brat. What was wrong with him this morning? The next words should have been didn’t ask permission, but in their place hung this resistant silence. “Then give me permission,” Ashenivir whispered. “Give me permission, Master, or teach me not to do it.” The air between them thrummed. Rizeth’s eyes darkened, and he urged it on as he felt a light brush of magic in the mark on the back of his neck. Yet still Rizeth only held him, and maybe he was building tension, but Ashenivir didn’t want to wait. He grabbed two fistfuls of Rizeth’s hair and pulled, hard.

stars

The new bed was a large, solid creation—dark wood, sensible and sturdy, the only adornment a series of interlocking moons and eight-pointed stars along the head and footboards. The oak had been his choice; the decoration Ashenivir’s.

blue

The ballroom was marginally more tolerable than the foyer. Cooling charms rippled in wave-patterns of blue and ice-white along one wall, and a handful of servants were in the process of painting similar magic onto the columns lining the room. They paused by one just as the final sigil was completed, and a wave of cold air rolled over them.

Obedience taglist: @foxboyclit @belovedviolence (ask to be +/-)

#writeblr #tag games #find the word #find the word tag #fanfiction #snippets #wip #obedience fic blogging #c: ashenivir zauvym #c: rizeth velkon’yss #c: catriona hanali #one thing about me is i will abuse online dictionaries of fictional languages to invent more titles for my fantasy d/s relationships #salen’cath is roughly ‘my cat’#bc he also calls her kitten and her name is catriona #also the boys choosing a bed together be still my heart

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