Bartosz Milewski on Nostr: Tannakian reconstruction is both profound and trivial. Consider a presheaf on C: ...

Tannakian reconstruction is both profound and trivial. Consider a presheaf on C:
𝐹:𝐶ᵒᵖ→𝑆𝑒𝑡 .
It's value at some object a is the set Fa. Now vary F but keep a fixed. You get a functor from the category of presheaves to Set.
Φₐ:[𝐶ᵒᵖ,𝑆𝑒𝑡]→𝑆𝑒𝑡
Φₐ𝐹=𝐹𝑎
This is often called a fiber functor. Now fix another object b. What can you say about the set of natural transformations between two fiber functors?
\[ [[C^{op}, Set], Set] (\Phi_a, \Phi_b) \cong \int_F (F a \to F b)\]
The end here is taken over all possible presheaves. It's a gigantic product, so it's enough that one of its components is empty to make the whole end empty.

So what prevents us from picking a presheaf that's non-empty over a and empty over b? There is no function from a non-empty set to an empty one, so the whole end would end up empty. One bad apple spoils the whole batch.

Except that we can't do it if there is a morphism 𝑓:𝑎→𝑏 in C. Functoriality of F means that there always is a function
𝐹𝑓:𝐹𝑎→𝐹𝑏
In fact, it can be shown (using a double Yoneda trick) that there is a one to one correspondence between the homset C(a, b) and the set of natural transformations between fiber functors:
\[ \int_F (F a \to F b) \cong C(a, b)\]
So that's the Tannakian reconstruction in the categorical language. It's simplest application is for C a single-object category, i.e. a monoid. We can reconstruct the monoid (the hom-set) from its representations (the presheaves).