She hangs. Threads threaded through her veins and out of her soles, then joined bare some paces and finally tied to the chandelier, from which candlewax dripping and following the curves and paths and arches, finds its way down to the central nipple pointed down, trailing down that bundle of threads like raindrops on a car window and thinning it little by little by
       She hangs, crucified upsidedown like a bat, arms spread like arms, and the sleeves of her blouse like wingfilm, breeze through open windows speaking through flimflam filmflap. Laid up and swung out and strung down. Deathly dangling, dark circumferences of socket holding ballish greening eyes. Swinging to, fro, to, fro, implicitly shouting for Mama Wind to push, and that classic coincidence of apathetic nature generously responding to unnatural call.
   Her shadow hangs on the Western Wall, innocence swindled. Urine yanked out. Unrivaled. She hangs, dead center in the room, woman of the hour. A statue towering from above, a ceiling mural reaching out into the room and scooping out reality. Prorated and forever contemporary. She is wading in the air; a freeform pendulum counting something unknown. A private countdown as the year reaches its own end.
     She hangs ahoy, ankles shackled and head lifted from the Earth. Pulled out like an onion and partly peeled back. A treasure chest sprung out of the shore. Skin picked and shedding like sand. Decaying in dim shapes of light and shadow, dignity pillaged, risen above mundanity yet forever faithful to it.
    She hangs yes she does tither, hither, like an inverted rocking horse, left-right orientation matching the face-to-face voyeur. Cadaverous inversion to straight life, whence true love. Spoilt love. That bloody, roachy corpse comfortably exposed as if mummified, as if protected from public view by a gently rocking warm cocoon, as if paused in the middle of some transformation whose intermediate steps are as grotesque as the finale is surely beautiful.
        She hangs, hair sickly, coming off. Half a nostril missing. Tongue licking upper lip. Grayfaced youth, arms squandered, sinking more trying to reach the depths beyond the sullied floor indeed to fly down into deep heights never reached before if not to collapse into a intestinal voodoo-jointed mess upon hitting the floor when the last thread loses its grip. And under crusty forehead caved in cheeks through caustic lips, dead but alive, a voice within a voice sounds: some alien kernel within speaking out of that thin hole:

1 Left on right

 
Let's try and figure out

To define ϕ, I need to define, for every open set U ⊂ X,

(1)

Then simply reading out the definitions above, I can unwrap the lhs as

f-1f * (U)	= lim V ⊃f(U)f* (V )
	= lim V ⊃f(U) (f-1(V ))

So (2) is

(2)

ALRIGHT, THAT LEFT HAND SIDE LOOKS MESSY AS HELL. Let's denote the restriction maps of f^-1f_* as ρ′. And the restriction maps of f_* and as ρ (denoting them the same since the former inherits the latter). The question is: What to elements in the left set looks like? It's a direct limit, so they look like this:

< V,s >

s.t. V ⊃ f(U),s ∈ (f-1(V ))

with the equivalence relation

"Dude, what the fuck is this shit"

I...It makes sense if you think abt it.... The equivalence relation is: GIVEN AN OPEN SUPERSET OF f(U) ⊂ Y (since f(U) isn't guaranteed to be open itself), PULL BACK TO AN OPEN SET IN X, RESTRICT, AND COMPARE.

Now, the major insight here as far as defining ϕ(U) is that s ∈ (f^-1(V )), and

So we can restrict s to an element of (U). HENCE, THE NATURALLLLL DEFINITION OF OUR MAP IS

< V,s >

ρf-1(V ),U(s)

 

"Alright, great. Now, is this map well-defined?"

Ha... Hahaha... Hahahahaha... Mua.. Muahahaha. MUAHAHAHAHAHAHAHAHA. MUAHAHAHAHAHAHAHAHAHAHAH. MUAHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHA. MUAHAHAHAHAHHAAHAHAAAAAAAAAHAHA AHAHAHHHHHHHAHAAAAAAAH HAAAAAAAAAAAAAAAAAAAAAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAH

1.1 Cute and valid?

Answer: FUCK IF I KNOW. Well, okay, I actually DO KNOW that it IS... But I CAN'T PROVE IT!!! Look:

ϕ(< V,s >)	= ϕ(< W,t >)
ρf-1(V ),U(s)	= ρf-1(W),U(s)

With this info, I'm supposed to prove the equality

< V,s >

< W,t >

holds, right? WELL..... I CAN'T. Because in order for that equality to hold, looking back at (5), I need some set T ⊃ f(U) in Y such that

Now, look... I know when restricted to U, s = t, but every set f^-1(T) contains U, and in particular may not equal U. I need a set larger than U in which s = t (and furthermore is inversely mapped onto from an open set in Y ).... HOW THE FUCK?

I'm moving on

2 Right on Left

 
Okay, this one is less of a pain in the butt. We want a NATCHURAL map

I.e. we need to make, given open V ⊂ Y ,

So the side that's needs to be unraveled this time is the RHS:

f*f-1 (V )	= f-1 (f-1(V ))
	= lim W⊃f(f-1(V )) (W)

Okay, the key point here is V itself is one of those Ws, since V ⊃ f(f^-1(V )), so our map can obviously just be

NO CUTE AND VALID CHECK NEEDED. YAY.

3 The moment you've all been waiting for: Adjointness

 
Alright, I need a NATCHURAL bijection

Okay, what I decided to do after getting brainfucked trying to think about this was, let's break it up. Start with the simpler problem:

3.1 my new creation: the f_* operator on morphisms

 
Let's say , , are sheaves on X, f : X → Y is continuous, and suppose h : →. I would like to induce a map

i.e. for any open set V ⊂ Y , I need to define

f*h(V ) : f* (V )	→ f* (V )
i.e. f*h(V ) : (f-1(V ))	→ (f-1(V ))

at which point, the definition becomes obvious (NATURAL!):

3.2 inducing le map using the f_* operator

 
So, basically, now given h ∈ Hom_X(f^-1 , ), i.e.

h : f-1

→

I can induce a map

f*h : f*f-1

→ f *

and NOW to get the desired element of Hom_Y ( ,f_* ) all I have to do is right compose with the previously created ψ. Behold:

 

"......................................................"

Err... yea. I tried to use tikz-cd, and as you can see, tex4ht did not respond well do that...
Here, I'll give you the screenshot from my editor instead:

3.3 The f^-1 operator

 
Since I'm making a bijection, I'd like to make a map in the opposite direction. So...

Let , be sheaves on Y , and f : X → Y a cont map, and let h : →. Now I want to define a map

i.e. given open U ⊂ X, I'd like to make

f-1h(U) : f-1 (U)	→ f-1 (U)
i.e.f-1h(U) : lim V ⊃f(U) (V )	→ lim W⊃f(U) (W)

so the "obvious" NAAATURAL map is just

3.3.1 cute and valid?

Fuck you, I'm not checking

3.4 Inducing the inverse map using _f^-1

 
Now given h : → f_* , I can induce a map

f-1h : f-1

→ f-1f *

and then get the desired map by left composing with ϕ from above:

 

oofff.. that ain't it bros... that ain't it....

Well, here's a screenshot again of what it was SUPPOSED to be:





Done

3.5 Happy new year

 
Now we have to verify that we made a bijection; that yey maps to yay and viceversa. Seems like quite a pain. But the more you look, as all "natural" things are, it's borderline computational. The problem shall solve itself as long as you follow the path. Walk, and the contructions should construct themselves. Carry yourself by your ideal visions of yourself and fall unconsciously into the bliss of being, as the world collapses around you.

When the skies are dark, it enables you to cut out the extraneous stuff. It enables you to harvest that tinge of evil in you. You know what you have to do and why. You've known it for years. It's been inside you as long as you can remember.

Now's the year. All you have to do is keep your eyes on the prize and you'll be unstoppable. I know you can do this. Hence,

 



Welcome to 2022. It's here, whether you like it or not.