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).

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The paradox of the stone: "Could God create a stone so heavy that even he could not lift it?" was solved by Grothendieck by introducing an infinite tower of gods. The God of the n'th universe can only create stones that he can lift. But the God from the n+1 universe can create an unliftable stone for the God from the n'th universe, and so on.

Cell complex -- something you get when your friend can afford the newest version of the iPhone and you can't.

Also: an object in a category which is obtained by successively “gluing cells” via pushouts.

Democracy is like health: You don't appreciate it until you lose it.

The piñata effect: At the final moment of a black hole evaporation through Hawking radiation, all objects that were swallowed by it suddenly come back, thus restoring the conservation of information required by quantum mechanics.