Lately been falling asleep doing the #SplittingLemma in my head.

It is fun to be #CountingSheep by a simple diagram chase!

To recap, given the short exact sequence
0 → 𝐴 → 𝐵 → 𝐶 → 0 it splits if 𝐴 → 𝐵 is invertible at 𝐴 or 𝐵 → 𝐶 is invertible at 𝐶. In either equivalent case, 𝐵 ≅ 𝐴⊕𝐶.

The fun part is now to prove all of these equivalences in your head so you can fall asleep. For simplicity, let's assume an abelian category. Or just plain modules (kind of implied by the direct sum, I think, you get a partial splitting lemma in nonabelian groups).

Let's do 𝐴 → 𝐵 is invertible implies 𝐵 ≅ 𝐴⊕𝐶 !

Giving these morphisms names, let 𝑖: 𝐴 → 𝐵 and 𝑗: 𝐵 → 𝐶 and let's call the inverse 𝑝: 𝐵 → 𝐴. Need to check that 𝑏 ↦ (𝑝(𝑏), ,𝑗(𝑏)) is both mono and epi.

Let's begin with (𝑝(𝑏), ,𝑗(𝑏)) = (0,0) implies 𝑏=0. Since 𝑏 ∈ ker 𝑗, by exactness at 𝐵 we have that 𝑏 = 𝑖(𝑎) for some 𝑎. But then 0 = 𝑝(𝑏) = 𝑝(𝑖(𝑎)) = 𝑎 because 𝑝 is an inverse. Since 𝑖 is mono, 0 = 𝑖(0) = 𝑖(𝑎) = 𝑏.

Now we need to construct 𝑏 such that (𝑝(𝑏), 𝑗(𝑏))=(𝑎, 𝑐) for arbitrary 𝑎 and 𝑐. Since 𝑗 is epi, there exists 𝑏' such that 𝑗(𝑏') = 𝑐. But 𝑝(𝑏') might be any old thing! Let's patch up 𝑏' and consider 𝑏 := 𝑏' + 𝑖(𝑎−𝑝(𝑏')). By exactness at 𝐵, we also have that 𝑗(𝑏) = 𝑗(𝑏') = 𝑐. As a similar check, 𝑝(𝑏) = 𝑝(𝑏'+𝑖(𝑎−𝑝(𝑏'))) = 𝑝(𝑏')+𝑝(𝑖(𝑎−𝑝(𝑏')) but since 𝑝 is an inverse this is 𝑝(𝑏')+𝑎−𝑝(𝑏') = 𝑎, proving that 𝑏 has been constructed as desired.

🐑 A closer look at my little felt sheep from @teamspellbinders February release. Between the felt, the stitching and the 3D embossing folder, it's a very gentle and tactile card! It would be perfect for a baby shower.

🛒 Shop this card https://ldli.co/e/d7k720k

#countingsheep #spellbinderspaperarts #spellbindersdies #stampanddieset #justforewe #punsarefun #watercolorpainting #babyshower https://www.instagram.com/reel/DGA_2dToR9j/?utm_source=dlvr.it&utm_medium=mastodon

When I saw @teamspellbinders newCounting Sheep Stamp & Die Set, I immediately wanted to create the sheep out of felt. To make it extra fluffy, I doubled up the felt. Then I stitched around my watercolored background with white floss, to make it even more tactile.

More about this "just for ewe" card on my blog https://valleyoakpaper.com/2025/counting-sheep

#spellbinderspaperarts #neverstopmaking #spellbindersdies #countingsheep #stampanddieset #justforewe #watercolorcards #cardmaking https://www.instagram.com/p/DF5RXhuswja/?utm_source=dlvr.it&utm_medium=mastodon

You ever have insomnia so bad you start counting sheep, but then the sheep start counting you? Now I’m lying there, thinking, “Great, even the sheep are getting more sleep than I am!”

#Insomnia #CantSleep #SleepDeprived #CountingSheep
#LateNightThoughts

@Bruce I actually do this. Look at my toots under the #CountingSheep hashtag.

" https://punnypeak.com/sleep-puns/

I tried to count sheep 🐑 to fall asleep, but they kept jumping over the fence and interrupting my Zzz's 😴

#SleepJokes
#CountingSheep
#TroubleSleeping
#InsomniaHumor
#BedtimeLaughs
#SleeplessNights
#FenceJumpingSheep
#ZzzzStruggles
#LaughTillYouSleep
#ComedyDreams"

https://punnypeak.com/sheep-puns/ "
" Why was the sheep always tired? Because she had to keep counting herself to sleep! #sheep #tired #insomnia #countingsheep #baaadtime #sleeplessnights #woolyjoke #flockyeah #ewearewhatsheepeated #lambchops

#meme #countingsheep #tofu #vegan #humor

Five pillows and an indoor fountain: readers share their surprising sleep routines

https://www.theguardian.com/wellness/2023/nov/21/sleep-routine-how-people-fall-asleep?CMP=Share_iOSApp_Other #Sleep #SleepProblems #CountingSheep

"Sweet Dreams" teableau for 08/31/23

#Tea #Teableau #TeaCozy #TeaCosy #RockPainting #Chamomile #Lavender #HerbalTea #CountingSheep

@blueberry

Thanks. Now I finally know the true meaning of #tldr is The Lazy Data & Research. Great project, and more subversive than just #CountingSheep.

Exhausting day... so here's an appropriate choon.

From the 1991 Album Boing!

#Airhead
#boing
#CountingSheep
#MondayMusic

and with that, I'm off to bed at 8:20 in the evening... what a frickin lightweight I am.

https://www.youtube.com/watch?v=frNsD1ktDlk

Seriously cannot sleep, tonight.

Trying to force slumber rarely works.

Instead I’ll settle for staring at the night-light projected on my ceiling, perhaps? 🤷🏻‍♂️😅

It is mesmerising.

Doesn’t seem to impact the cats none… they’re sleeping blissfully, enjoying the comfort of my body heat on this coolish #Melbourne night. 🐾🐾

🐈🐈‍⬛

#narrm #victoria #Australia #makesmehappy #countingsheep #zzz

Even the fediverse isn’t perfect ;)
Woke in the night and found my way over here. I was in a bad mood already. I found I was seeing a lot of bleak posts from people. A lot of politics, a lot of misanthropy. Was it my mood that primed me to see it? It’s also worth noting one reason I was already feeling focused on misery is I’m a United Statesian. That’s enough of an explanation on a normal day. It’s been a very bad few days here. I think that’s a big part of why I was down and why I saw down posts from others (in English, mostly fellow USians). Now, after a few hours rest I am feeling better and I log on and see the usual mix of hopeful, creative, critical, etc.
Everything is everything, etc.

I initiated a wonderful activity as I was trying to return to sleep in the middle of the night, inspired by my love of Gertrude Stein’s Tender Buttons (a work I’ve been revisiting via @TenderButtons@botsin.space). I have played word games while falling asleep before but there has always been some structure to them. One favorite is to create a list of words in which one’s final letter becomes two’s first letter, and so on. There can be restrictions of type of word, world cities for example (London, Naples, Singapore…). It’s a fine word game. Possibly it’s overly active mentally to make it ideal for falling asleep. Last night I found myself producing a list of words based on sound or feeling only. I can’t say random but I don’t know where they came from. I simply allowed a stream of words to cross my mind, appear, disappear, with as little conscious thought as possible. I felt relaxed and fell asleep.

What variations on #countingSheep have others with #insomnia tried?

Counting Sheep

It's more effective to count all the ribs

#art #ascii #asciiart #bbs #computer #computerart #demoscene #digital #digitalart #retro #retroart #text #textart #textmode #textpunk #8bit #8bitart #sheep #skull #skeleton #occult #sleep #countingsheep #unfinished

Making Rockets Dull Again

https://venera.social/display/85a863ed-3963-bca5-0412-30c382864346

#MastoPrompt #SmallPoem

1, a place to start
2, keep it even on the ark
3, puts one in the middle
4, counts in the dark
5, brain is a beehive
6, hit me with a brick
7, is not really lucky
8, I give in to fate
9, not feeling so fine
10, it’s getting awfully late
11, easy on the eye
12, so tired I could cry
13, is awkward to say
14, it’s another day
15, many years ago
16, maybe a midnight snack
17, my feet feel #frozen
18, yawn attack
19, feel kinda cosy
20, I’m dozy
21, Zozzy
22,

#CountingSheep

I haven't done one of these in a while, but lately when I'm falling asleep, I have been doing some #matrices in my head.

A simple one has been the #Pascal matrix.

Let's do another episode of #CountingSheep!