GoKawiilTwo 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.
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.
In general it’s impossible to recover the structure of the source category by looking at only one such picture. But if the target category has enough resolution, and we superimpose all available pictures, we can recover the morphisms of the source category.
Fiber functors
A category with just the right resolution is the category of sets. Therefore we’ll be looking at functors in (for historical reasons, these are called co-presheaves). Such a functor maps objects to sets, and morphisms to functions.
When dealing with functors, we usually imagine varying objects and morphisms while keeping the functor constant. Here, we are interested in using the totality of all functors while keeping the objects constant. To every object we will associate a mapping from functors to sets, simply by applying each functor to this object :
This mapping is functorial. Indeed, a natural transformation between and is a family of functions . We define the action of on a natural transformation by taking its component .
is called a fiber functor. You may think of it as probing an object and, through morphisms, its immediate neighborhood.
Tannakian reconstruction
To probe a hom-set we’ll be looking at the set of functions under all possible functors .
... continue reading