[HoTT] Re: A definition of equivalence where the identity is a unit on both sides for composition

["alexr...@gmail.com" <alexrice73@gmail.com>]

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Hi Jasper,

I have also thought about these sort of structures a bit, and in fact this 
sort of "trick" of decomposing something of passing in an equality also 
works in a variety of situations where composition involves transitivity of 
equality. For example, you might be interested in Martin Escardo's version 
of this for equality <https://www.cs.bham.ac.uk/~mhe/yoneda/yoneda.html> 
which contains a fair bit of discussion on the topic. I have also done some 
work on this sort of idea related to groups which can be found here 
<https://alexarice.github.io/posts/sgtuf/Strict-Group-Theory-UF.html> and 
submitted a pull request to agda standard library with a similar idea to 
this for quasi-inverses <https://github.com/agda/agda-stdlib/pull/1156>. 

With respect to the questions at the end, I'm afraid I am not aware of any 
definition for which inverses are strict and I expect that (2) (at least in 
it's full generality) is not possible as this would imply that 
\infty-groupoids are equivalent to strict \infty-groupoids. For involution 
you could perhaps do a similar trick that is done in category theory 
libraries like agda-categories where you store the data for both 
directions. More precisely, if we let your definition be called biEquiv 
then you could let

A involutiveBiEquiv B = A biEquiv B x B biEquiv A

and then let inverting be given by swapping the order of the product. I 
believe this should maintain the nice compositional properties of biEquiv 
and adds involutive inverses for "free".

Hope some of this was useful/interesting.
Alex

On Sunday, 16 August 2020 at 20:08:47 UTC+1 jas...@cs.washington.edu wrote:

> I was thinking about various definitions of equivalence, and the various 
> equations we could ask them to satisfy up to definitional equality. (They 
> are of course all the same when looking at propositional equality.)
>
> Looking at composition, all the definitions of equivalence I have seen 
> before satisfy at most one of the two equations id o p = p or p o id = p 
> (for p : A equiv B and _o_ composition).
>
> By adapting the definition by bi-invertible maps, we can get a definition 
> of equivalence where both the above equations hold, and additionally we get 
> definitional associativity (p o q) o r = p o (q o r).
>
> -----
>
> Recall the bi-invertible map definition of equivalence is
> A equiv B =
>   (f : A -> B) x
>   (gl : B -> A) x
>   (gr : B -> A) x
>   (linv : forall a, gl(f(a)) = a) x
>   (rinv : forall b, f(gr(b)) = b)
> Then the identity is (id, id, id, refl, refl).
>
> When composing (f1, gl1, gr1, linv1, rinv1) with (f2, gl2, gr2, linv2, 
> rinv2),
> we can take f = f1 o f2, gl = gl2 o gl1, gr = gr2 o gr1, but linv and 
> rinv require path induction (essentially to compose instantiations of linv1 
> and linv2). Since path composition in an arbitrary type only satisfies one 
> of the two unit equations definitionally, I don't believe it is possible to 
> satisfy both id o p and p o id for this definition.
>
> However, we can modify the definitions of linv and rinv inspired by the 
> symmetric coinductive definition of equivalence (that A equiv B = (f : A -> 
> B) x (g : B -> A) x forall a b, (f a = b) equiv (a = g b)).
>
> So we take linv : forall a b, f a = b -> a = gl b, and rinv : forall a b, 
> a = gr b -> f a = b.
>
> (Notice that (b : B) x (f a = b) and (a : A) x (a = gr b) are 
> contractible, so this definition is equivalent to the bi-invertible map 
> definition, and thus a real definition of equivalence.)
>
> Then for the identity, we take linv = rinv = id.
> Thus for composition, we can take linv = linv2 o linv1 and rinv = rinv1 o 
> rinv2, and exploit the fact that the identity function is definitionally a 
> two-sided unit for function composition.
>
> Q.E.D.
>
> -----
>
> Has anyone seen this definition or other definitions with this property 
> used before?
>
> This definition does not behave particularly nicely with respect to 
> inversion. I know a few definitions with definitional involutive inversion, 
> and most definitions seem to satisfy id^-1 = id, but I don't think I know 
> any definitions where p^-1 o p = id or p o p^-1 = id.
>
> There is such a wide variety of possible choices with respect to the 
> definition of equivalence, I am interested in what additional properties we 
> can ask of it to narrow down the scope of a "good" definition, and what 
> unavoidable trade offs exist.
> Some good properties I am thinking about are:
> 1) Decomposition into (f : A -> B) x isEquiv f
> 2) Definitional groupoid equations.
> Currently (1) seems to be incompatible with involutive inversion, but as 
> shown here we can have both (1) and definitional equations for everything 
> except inversion. Can we be more greedy?
>
> Sincerely,
> - Jasper Hugunin
>

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