1st grade mathematics work sheets and my mother's mathematics teaching style. Mathematics will not be so terrible, since it seems that parents are interested in preparing their little ones for mathematics before school age.
More from Arthouse-Kuklite
More from Maths
In light of my tweet thread about the category of finite sets and commutative monoids (https://t.co/jnY0wZZbxq), I thought I might try to say what the analogue is for braided monoidal things (although much of this is still somewhat hypothetical).
It's also just kind of a cool combinatorial structure! I've been talking to @CreeepyJoe about this lately, as well as @grassmannian.
The first thing you have to know is that, in a braided monoidal category you can still have commutative monoids. Since a braided monoidal category C has a "twist" map for every object β(x):x⊗x→x⊗x, if x is a monoid you can ask for the following diagram to commute:
Remember that being symmetric monoidal just means that if you take the twist map above and do it twice, you get the identity map, but braided monoidal doesn't mean that. But it's okay! You can still define commutative monoids here.
But so anyway, we can talk about commutative monoids in braided monoidal categories.
So okay, here's a thread on the category of finite sets and a way in which it controls algebraic structure in symmetric monoidal categories. I think it's some really pretty stuff.
— Jonathan Beardsley (@JBeardsleyMath) December 6, 2020
It's also just kind of a cool combinatorial structure! I've been talking to @CreeepyJoe about this lately, as well as @grassmannian.
The first thing you have to know is that, in a braided monoidal category you can still have commutative monoids. Since a braided monoidal category C has a "twist" map for every object β(x):x⊗x→x⊗x, if x is a monoid you can ask for the following diagram to commute:
Remember that being symmetric monoidal just means that if you take the twist map above and do it twice, you get the identity map, but braided monoidal doesn't mean that. But it's okay! You can still define commutative monoids here.
But so anyway, we can talk about commutative monoids in braided monoidal categories.
Loops and their multiplication groups
A thread in 15 parts
(0/15)
Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.
(1/15)
Note that we don't assume associativity of *!
A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
(2/15)
Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples
For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.
(4/15)
A thread in 15 parts
(0/15)
Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.
(1/15)
Note that we don't assume associativity of *!
A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
(2/15)
Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples
Rethinking Vector Addition
— Michael Kinyon (@ProfKinyon) December 1, 2020
or
How I Learned to Stop Worrying and Love Nonassociativity
A thread in 29 tweets
(0/28)
For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.
(4/15)
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I hate when I learn something new (to me) & stunning about the Jeff Epstein network (h/t MoodyKnowsNada.)
Where to begin?
So our new Secretary of State Anthony Blinken's stepfather, Samuel Pisar, was "longtime lawyer and confidant of...Robert Maxwell," Ghislaine Maxwell's Dad.
"Pisar was one of the last people to speak to Maxwell, by phone, probably an hour before the chairman of Mirror Group Newspapers fell off his luxury yacht the Lady Ghislaine on 5 November, 1991." https://t.co/DAEgchNyTP
OK, so that's just a coincidence. Moving on, Anthony Blinken "attended the prestigious Dalton School in New York City"...wait, what? https://t.co/DnE6AvHmJg
Dalton School...Dalton School...rings a
Oh that's right.
The dad of the U.S. Attorney General under both George W. Bush & Donald Trump, William Barr, was headmaster of the Dalton School.
Donald Barr was also quite a
I'm not going to even mention that Blinken's stepdad Sam Pisar's name was in Epstein's "black book."
Lots of names in that book. I mean, for example, Cuomo, Trump, Clinton, Prince Andrew, Bill Cosby, Woody Allen - all in that book, and their reputations are spotless.
Where to begin?
So our new Secretary of State Anthony Blinken's stepfather, Samuel Pisar, was "longtime lawyer and confidant of...Robert Maxwell," Ghislaine Maxwell's Dad.
"Pisar was one of the last people to speak to Maxwell, by phone, probably an hour before the chairman of Mirror Group Newspapers fell off his luxury yacht the Lady Ghislaine on 5 November, 1991." https://t.co/DAEgchNyTP
OK, so that's just a coincidence. Moving on, Anthony Blinken "attended the prestigious Dalton School in New York City"...wait, what? https://t.co/DnE6AvHmJg
Dalton School...Dalton School...rings a
Oh that's right.
The dad of the U.S. Attorney General under both George W. Bush & Donald Trump, William Barr, was headmaster of the Dalton School.
Donald Barr was also quite a
Donald Barr had a way with words. pic.twitter.com/JdRBwXPhJn
— Rudy Havenstein, listening to Nas all day. (@RudyHavenstein) September 17, 2020
I'm not going to even mention that Blinken's stepdad Sam Pisar's name was in Epstein's "black book."
Lots of names in that book. I mean, for example, Cuomo, Trump, Clinton, Prince Andrew, Bill Cosby, Woody Allen - all in that book, and their reputations are spotless.