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.
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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.
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Stan Lee’s fictional superheroes lived in the real New York. Here’s where they lived, and why. https://t.co/oV1IGGN8R6
Stan Lee, who died Monday at 95, was born in Manhattan and graduated from DeWitt Clinton High School in the Bronx. His pulp-fiction heroes have come to define much of popular culture in the early 21st century.
Tying Marvel’s stable of pulp-fiction heroes to a real place — New York — served a counterbalance to the sometimes gravity-challenged action and the improbability of the stories. That was just what Stan Lee wanted. https://t.co/rDosqzpP8i
The New York universe hooked readers. And the artists drew what they were familiar with, which made the Marvel universe authentic-looking, down to the water towers atop many of the buildings. https://t.co/rDosqzpP8i
The Avengers Mansion was a Beaux-Arts palace. Fans know it as 890 Fifth Avenue. The Frick Collection, which now occupies the place, uses the address of the front door: 1 East 70th Street.
Stan Lee, who died Monday at 95, was born in Manhattan and graduated from DeWitt Clinton High School in the Bronx. His pulp-fiction heroes have come to define much of popular culture in the early 21st century.
Tying Marvel’s stable of pulp-fiction heroes to a real place — New York — served a counterbalance to the sometimes gravity-challenged action and the improbability of the stories. That was just what Stan Lee wanted. https://t.co/rDosqzpP8i
The New York universe hooked readers. And the artists drew what they were familiar with, which made the Marvel universe authentic-looking, down to the water towers atop many of the buildings. https://t.co/rDosqzpP8i
The Avengers Mansion was a Beaux-Arts palace. Fans know it as 890 Fifth Avenue. The Frick Collection, which now occupies the place, uses the address of the front door: 1 East 70th Street.
To people who are under the impression that you can get rich quickly by working on an app, here are the stats for https://t.co/az8F12pf02
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I have spent 1.5 months on this app. You can make more $ in 2 days.
🤷♂️
I'm still happy that I launched a paid app bcs it involved extra work:
- backend for processing payments (+ permissions, webhooks, etc)
- integration with payment processor
- UI for license activation in Electron
- machine activation limit
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These things seemed super scary at first. I always thought it was way too much work and something would break. But I'm glad I persisted. So far the only problem I have is that mailgun is not delivering the license keys to certain domains like https://t.co/6Bqn0FUYXo etc. 👌
omg I just realized that me . com is an Apple domain, of course something wouldn't work with these dicks
📈 ~12000 vistis
☑️ 109 transactions
💰 353€ profit (285 after tax)
I have spent 1.5 months on this app. You can make more $ in 2 days.
🤷♂️
I'm still happy that I launched a paid app bcs it involved extra work:
- backend for processing payments (+ permissions, webhooks, etc)
- integration with payment processor
- UI for license activation in Electron
- machine activation limit
- autoupdates
- mailgun emails
etc.
These things seemed super scary at first. I always thought it was way too much work and something would break. But I'm glad I persisted. So far the only problem I have is that mailgun is not delivering the license keys to certain domains like https://t.co/6Bqn0FUYXo etc. 👌
omg I just realized that me . com is an Apple domain, of course something wouldn't work with these dicks
