"equivalent" concept

Hi!

I would like to comment that I feel very atracted by this project. I read this project several times to little by little understand better the concepts that are written. However the concept of equivalence is confusing for me. The definition is defined by example and, then, is used on definitions of algebras. For example:

### Functor

1. `u.map(a => a)` is equivalent to `u` (identity)
2. `u.map(x => f(g(x)))` is equivalent to `u.map(g).map(f)` (composition)

But maybe, the type that implements Functor is not any of this type examples. Then, is it not better define equivalence in terms of Setoid that defines equals method?

I mean:

### Functor

Functor implements the Setoid specification

1. `u.map(a => a).equals(u)` (identity)
2. `u.map(x => f(g(x))).equals(u.map(g).map(f))` (composition)

Cheers!

[bumbleblym]

You can create a legal setoid that has different behavior for ‘equal’ values.

Even(2).equals(Even(4)) // true
Even(2).reduce((acc, x) => acc + x, 0) // 2
Even(4).reduce((acc, x) => acc + x, 0) // 4

[xaviervia]

I think this might relate to the problem of what equality is in general terms. “Equivalent” as used in the laws seems to me to be referring that the fact that the laws expect the equivalence to be “up to isomorphism”.

Defining an equality that works for laws (that is, not just for hard comparisons within the JavaScript language, but in general for the underlying mathematical theory that underpins them) is a problem that is probably outside the scope of Fantasy Land. If I understand this correctly, the problem of equivalence and isomorphisms is what drives Homotopy Type Theory and the Univalent Foundations.

So to recap, I think the meaning of “equivalent” when used in the context of the laws is different than the meaning of the “equals” function regarding the Setoid algebra as implemented within Fantasy Land. “equals” is in this case narrowly defined for the context of JavaScript, which might not have the expressivity to describe the concept of “equivalence”.