IT'S CHRISTMAS EVE BITCHES. YOU KNOW WHAT THAT MEANS. HYPE HYPE HYPE HYPE HYPE HYPE HYPE HYPE. AWWWW YEA MOTHERFUCKERS, YOU KNOW IT. IT'S YOUR FAVORITE. IT'S TIME FOR ANOTHER YEAR OF 性の６時間.

Hahahaha. It's funny because you're alone. HAHAHAHAHA LOOK AT YOU, READER. EVERYONE'S HAVING SEX EXCEPT YOU. YOU'RE READING THIS BLOG INSTEAD OF PARTICIPATING IN THE WORLD'S GREATEST ANNUAL ORGYYYYYYYYYYY wait actually that's the Great Barrier Reef party.

"Umm, that's great and all but why did you skip to 1.17?????" Well, I've said before that I shouldn't overdo the amount of exercises per section I do. And this section has... 22 exercises. Oh no! But when I looked closer it turns out that the first half of them are just defining and establishing basic objects in the category of sheaves, like direct sums and stuff. Okay, that's great and all but this is less important than... you know... actually fucking doing things with sheaves, which the second half is about. I can deal with those constructions as they cum.


HO HO HO HO HO HO HO HO HO HO HO HO HO HO HO. ARE YOU LONELY READER? WELL HOLEINMYHEART IS COMING TO TOWN. TO FUCK YOU. I'M COMING TO YOUR HOUSE TO FUCK YOU IN THE ASSHOLE. I'M GONNA KISS YOU AND STROKE YOUR HAIR AND LICK YOUR NIPPLES AND YOUR TOES AND YOUR GENITALS YES I'M GONNA SUCTION SUCK YOUR COCK AND STICK IT IN MY EYE ("Dude, why do you keep making jokes about eye socket sex on your website? It was mildly funny the first time but it's starting to get genuinely creepy. It's almost like you have some kind of obsession about it and you can't help but repeatedly bring it up as a 'joke' as a way of") WE'RE GOING TO HAVE A LOT OF ANAL SEX. *pulls your pajamas off and sticks my cock into your butthole and starts thrusting extremely rapidly smiling while you moan in both pain and pleasure tears fall out of your eyes in both sentimentality and sensuality while in the background plays your favorite xmas song* *After I cum inside you I wipe your butt with a tissue and kiss you on the lips and complimenting you and whispering i love you into your ear and kiss you on the cheek and i wipe my forehead in exhaustion* Whew its been a long time since I've sexually assaulted the reader.

The first part of this exercise, btw, is pretty trivial. I'm gonna denote Z = {P}^- for simplicity. If Q ∈ Z, then by the definition of closure, for every neighborhood V of Q, P ∈ V , which means the sections of each neighborhood of Q is A. So the stalk of Q is A. If Q Z, it's the opposite situation. See? Ok? EZ. Bit too Englishy but blejbljba.

Uhhhh he more complex part is the second part about the constant sheaf. First of all, here's the definition of the direct image:



Now I'm gonna go ahead and denote the constant sheaf here as since that's less confusing. So our map is i : Z → X and we're lookin' at the sheaf i_* . ALRIGHTY. So given open U ⊂ X, we have

i* (U)	= (i-1(U))
	= (U ∩ Z)

okay. SO if U ∩ Z = ∅, then P U, and also

i* (U)	= i* (∅)
	= 0	(a basic property of presheaves)

otherwise, U ∩Z≠∅, which means ∃Q ∈ Z such that Q ∈ U. But Z is the closure of P, and U is a neighborhood of Q which means that Q has to also contain P. So this is the P ∈ U case, and we know we have to get (U ∩ Z) = A. DO WE? Well, note that if we prove that U ∩ Z is connected, we're done. OK. So since the singleton {P} is connected, so is its closure Z, and so is the set U ∩ Z "in between" them (SAUCE)

AND THAT'S TODAY'S. I'll make another post on NYE. MERRY CHRISTMAS.