Why do Calculus of Construction based languages use Setoids so much?
[The answers by Lia-yao Xia and Floris van Doorn are both excellent, so I will try to augment them with additional information.]
Another view is that quotients, while used a lot in classical mathematics, are perhaps not so great after all. Not taking quotients but sticking to Groupoids is exactly where non-commutative geometry starts from! It teaches us that some quotients are incredibly badly behaved, and the last thing we want to do (in those cases!) is to actually quotient. But that the thing itself is not so bad, even quite good, if you treat it using the 'right' tools.
It is arguably also deeply embedded in all of category theory, where one doesn't quotient out equivalent objects. Taking of 'skeletons' in category theory is regarded to be in bad taste. The same is true of strictness, and a host of other things, all of which boil down to trying to squish things down that are better left as they are, as they do no harm at all. We're just used to wanting 'uniqueness' to be reflected in our representations - something we should just get over.
Setoid hell arises not because some coherences must be proven (you need to prove them to show you have a proper equivalence, and again whenever you define functions on raw representations instead of on the quotiented version). It arises when you're forced to prove these coherences again and again when defining functions that can't possibly "go wrong". So Setoid hell is actually caused by a failure to provide proper abstraction mechanisms.
In other words, if you're doing sufficient simple mathematics, where quotients are well-behaved, then there should be some automation that lets you work with that smoothly. Currently, in type theory, working out exactly what that could look like, is ongoing research. Floris' answer outlines well what one pitfall is: at some point, you give up that computations will be well-behaved, and from then on, are forced to do everything via proofs.
As Li-yao Xia's answer describes, setoids are used when we don't have or don't want to use quotients.
In the HoTT book and in Lean quotients are (basically) axiomatized. One difference between Lean and the HoTT book is that the latter has many more higher inductive types, and Lean only has quotients and (regular) inductive types. If we just restrict our attention to quotients (set quotients in the HoTT book), it works the same in Lean and in Book HoTT. In this case you just postulate that given a type A and an equivalence R on A you have a quotient Q, and a function [-] : A → Q with the property ∀ x y : A, R x y → [x] = [y]. It comes with the following elimination principle: to construct a function g : Q → X for some type X (or hSet X in HoTT) we need a function f : A → X such that we can prove ∀ x y : A, R x y → f x = f y. This comes with the computation rule, that states ∀ x : A, g [x] ≡ f x (this is a definitional equality in both Lean and Book HoTT).
The main disadvantage of this quotient is that it breaks canonicity. Canonicity states that every closed term (that is, a term without free variables) in (say) the natural numbers normalizes to either zero or the successor of some natural number. The reason that this quotient breaks canonicity is that we can apply the elimination principle for = to the new equalities in a quotient, and a term like that will not reduce. In Lean the opinion is that this doesn't matter, since in all cases we care about we can still prove an equality, even though it might not be a definitional equality.
Cubical type theory has a fancy way to work with quotients while retaining canonicity. In CTT equality works differently, and this means that higher inductive types can be introduced while keeping canonicity. Potential disadvantages of CTT are that the type theory is a lot more complicated, and that you have to embrace HoTT (and in particular give up on proof irrelevance).
Ideally one would certainly like to be able to treat arbitrary equivalence relations as Leibniz equality (eq), enabling rewriting in arbitrary contexts. That means to define the quotient of a type by an equivalence relation.
I'm not an expert on the topic, but I've been wondering the same for a while, and I think the reliance on setoids stems from the fact that quotients are still a poorly understood concept in type theory.
- Setoid Hell is where we're stuck when we don't have/want quotient types.
- We can axiomatize quotient types, I believe (I could be mistaken) that's what Lean does.
- We can develop a type theory which can naturally express quotients, that's what HoTT/Cubical TT do with higher inductive types.
Furthermore, quotient types (or my naive imagination of them) force us to package programs and proofs together in a perhaps less-than-ideal way: a function between two quotient types is a plain function together with a proof that it respects the underlying equivalence relation. While one can technically do that, this interleaving of programming and proving is arguably indesirable because it makes programs unreadable: one often seeks to either keep programs and proofs in two completely separate worlds (so that mandates setoids, keeping types separate from their equivalence relations), or to change some representations so the program and the proof are one and the same entity (so we might not even need to explicitly reason about equivalences in the first place).