Algebra 1 and pre-algebra worksheets are something that all would love to have been able to dominate in high school. Sometimes they do not dominate it there and that's when it could be a bit difficult.
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.
OK, I may be guilty of a DoS attack attempt on mathematicians' brains here, so lest anyone waste too much precious brain time decoding this deliberately cryptic statement, let me do it for you. •1/15
First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15
Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15
Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15
… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
\u2200x.\u2200y.((\u2200z.((z\u2208x) \u21d2 ((\u2200t.((t\u2208x) \u21d2 (t\u2208z) \u21d2 (t\u2208y)))) \u21d2 (z\u2208y))) \u21d2 (\u2200z.((z\u2208x) \u21d2 (z\u2208y))))
— Gro-Tsen (@gro_tsen) February 12, 2021
First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15
Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15
Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15
… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
You May Also Like
1. Project 1742 (EcoHealth/DTRA)
Risks of bat-borne zoonotic diseases in Western Asia
Duration: 24/10/2018-23 /10/2019
Funding: $71,500
@dgaytandzhieva
https://t.co/680CdD8uug
2. Bat Virus Database
Access to the database is limited only to those scientists participating in our ‘Bats and Coronaviruses’ project
Our intention is to eventually open up this database to the larger scientific community
https://t.co/mPn7b9HM48
3. EcoHealth Alliance & DTRA Asking for Trouble
One Health research project focused on characterizing bat diversity, bat coronavirus diversity and the risk of bat-borne zoonotic disease emergence in the region.
https://t.co/u6aUeWBGEN
4. Phelps, Olival, Epstein, Karesh - EcoHealth/DTRA
5, Methods and Expected Outcomes
(Unexpected Outcome = New Coronavirus Pandemic)
Risks of bat-borne zoonotic diseases in Western Asia
Duration: 24/10/2018-23 /10/2019
Funding: $71,500
@dgaytandzhieva
https://t.co/680CdD8uug
2. Bat Virus Database
Access to the database is limited only to those scientists participating in our ‘Bats and Coronaviruses’ project
Our intention is to eventually open up this database to the larger scientific community
https://t.co/mPn7b9HM48
3. EcoHealth Alliance & DTRA Asking for Trouble
One Health research project focused on characterizing bat diversity, bat coronavirus diversity and the risk of bat-borne zoonotic disease emergence in the region.
https://t.co/u6aUeWBGEN
4. Phelps, Olival, Epstein, Karesh - EcoHealth/DTRA
5, Methods and Expected Outcomes
(Unexpected Outcome = New Coronavirus Pandemic)
Oh my Goodness!!!
I might have a panic attack due to excitement!!
Read this thread to the end...I just had an epiphany and my mind is blown. Actually, more than blown. More like OBLITERATED! This is the thing! This is the thing that will blow the entire thing out of the water!
Has this man been concealing his true identity?
Is this man a supposed 'dead' Seal Team Six soldier?
Witness protection to be kept safe until the right moment when all will be revealed?!
Who ELSE is alive that may have faked their death/gone into witness protection?
Were "golden tickets" inside the envelopes??
Are these "golden tickets" going to lead to their ultimate undoing?
Review crumbs on the board re: 'gold'.
#SEALTeam6 Trump re-tweeted this.
I might have a panic attack due to excitement!!
Read this thread to the end...I just had an epiphany and my mind is blown. Actually, more than blown. More like OBLITERATED! This is the thing! This is the thing that will blow the entire thing out of the water!
Tik Tok pic.twitter.com/8X3oMxvncP
— Scotty Mar10 (@Allenma15086871) December 29, 2020
Has this man been concealing his true identity?
Is this man a supposed 'dead' Seal Team Six soldier?
Witness protection to be kept safe until the right moment when all will be revealed?!
Who ELSE is alive that may have faked their death/gone into witness protection?
Were "golden tickets" inside the envelopes??
Are these "golden tickets" going to lead to their ultimate undoing?
Review crumbs on the board re: 'gold'.
#SEALTeam6 Trump re-tweeted this.
