You are right to think it is puzzling! I am biased but I am of the opinion there are very deep phenomena lurking under these seemingly unimportant oddities.
To the a first degree the answer mostly boils down to the assymmetry in Set, the category of sets.
Set certainly doesn’t treat limits and colimits the same, so concepts built on top of it (like limits and colimits for ordinary categories) inherit its assymmetry.
As a very concrete instance of the assymmetry in Set:
The initial object 0 in Set is the empty set, which has a unique map to any other set (the empty map).
The terminal object 1 in set is the set containing one element. Every set X has a unique map to 1.
What about the other way around? Given a set X, what about the maps 1-> X? They exactly correspond to elements x of X !
Cool! Then what about maps X → 0 ?
Ah. Not so interesting.
There are none, unless X=0 is empty.
As pointed out by tailcalled, the opposite category of Set can be described as the category of complete atomic boolean algebras, a quite interesting category unto itself… but not equivalent to Set.
Cool, that’s all very good but it doesn’t explain why Set was assymetric in the first place.
Is this intrinsic? There are certainly natural categories that are equivalent to their opposites (the category of relations is a good examples).
To really understand what’s going on here we need to dig deeper. We need to step outside the traditional framework of set theory, and re-examine the foundations of type theory. If we do, we will find that even the very notion of category itself has an underlying bias, a broken duality. To be continued...
I think exponentials and coexponentials are relevant here, since they are good at shuffling things back and forth between the sides of morphisms, which matters for limits and colimits as they are adjunctions (and a particularly nice kind of adjunction, at that).
You are right to think it is puzzling! I am biased but I am of the opinion there are very deep phenomena lurking under these seemingly unimportant oddities.
To the a first degree the answer mostly boils down to the assymmetry in Set, the category of sets. Set certainly doesn’t treat limits and colimits the same, so concepts built on top of it (like limits and colimits for ordinary categories) inherit its assymmetry.
As a very concrete instance of the assymmetry in Set: The initial object 0 in Set is the empty set, which has a unique map to any other set (the empty map). The terminal object 1 in set is the set containing one element. Every set X has a unique map to 1.
What about the other way around? Given a set X, what about the maps 1-> X? They exactly correspond to elements x of X ! Cool! Then what about maps X → 0 ? Ah. Not so interesting. There are none, unless X=0 is empty.
As pointed out by tailcalled, the opposite category of Set can be described as the category of complete atomic boolean algebras, a quite interesting category unto itself… but not equivalent to Set.
Cool, that’s all very good but it doesn’t explain why Set was assymetric in the first place.
Is this intrinsic? There are certainly natural categories that are equivalent to their opposites (the category of relations is a good examples).
To really understand what’s going on here we need to dig deeper. We need to step outside the traditional framework of set theory, and re-examine the foundations of type theory. If we do, we will find that even the very notion of category itself has an underlying bias, a broken duality. To be continued...
This is an interesting post and I hope that you’ll continue it.
I think exponentials and coexponentials are relevant here, since they are good at shuffling things back and forth between the sides of morphisms, which matters for limits and colimits as they are adjunctions (and a particularly nice kind of adjunction, at that).