I admit I didn't follow this as much as I'd like to, and I wish it had some discussion of how this maps onto traditional signal processing frameworks in statistics and information theory, if it does.
Depending on your mathematical background, it may be easier to get a grasp on Tannaka duality, of which I assume, but don't know, this may usefully be viewed as a (possibly too) vast generalization.
>> Two friends, Alice and Bob, live in the same city, but on the opposite sides of a wide river. Every night, Bob looks at the lights on the other side and tries to guess, which one belongs to Alice. They come up with a clever arrangement: Alice will turn on her lights for 10 minutes every night at 10 p.m. Every night Bob will take a long-exposure photo at the pre-arranged time. At the end of the year, Bob will superimpose all the photos, and hopefully the only bright spot will be Alice’s window. This is Tannakian reconstruction in a nutshell.
Oh. That's cool. I totally get it. Simple!
>> A functor produces a picture of one category inside another. It’s a potentially lossy encoding, but it always preserves the structure of the source. If there is a connection (morphism) between two objects in the source category, there will always be a connection between their images in the target category.
I admit I didn't follow this as much as I'd like to, and I wish it had some discussion of how this maps onto traditional signal processing frameworks in statistics and information theory, if it does.
Depending on your mathematical background, it may be easier to get a grasp on Tannaka duality, of which I assume, but don't know, this may usefully be viewed as a (possibly too) vast generalization.
https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality
>> Two friends, Alice and Bob, live in the same city, but on the opposite sides of a wide river. Every night, Bob looks at the lights on the other side and tries to guess, which one belongs to Alice. They come up with a clever arrangement: Alice will turn on her lights for 10 minutes every night at 10 p.m. Every night Bob will take a long-exposure photo at the pre-arranged time. At the end of the year, Bob will superimpose all the photos, and hopefully the only bright spot will be Alice’s window. This is Tannakian reconstruction in a nutshell.
Oh. That's cool. I totally get it. Simple!
>> A functor produces a picture of one category inside another. It’s a potentially lossy encoding, but it always preserves the structure of the source. If there is a connection (morphism) between two objects in the source category, there will always be a connection between their images in the target category.
Whaat the fffff...