thank you Matt Oliveri.

the substitutions you used above explains my examples perfectly.
I also found a bug in my second examples :

f   : (A, t1, t2) -> (A, t, t3, t4)
g   : (B, t3, t4) -> (B, t6, t7)

(unify)
B = (A, t)

f   : (A, t1, t2) -> (A, t, t3, t4)
g   : (A, t, t3, t4) -> (A, t, t6, t7)

f;g : (A, t1, t2) -> (A, t, t6, t7)

(shorten)
f;g : (t1, t2) -> (t, t6, t7)

(but I mistakenly wrote)
f;g : (t, t1, t2) -> (t, t6, t7)

---

I will learn more about polycategory and polymorphic,
and try to use them to explain dependent type system.

thank you again :)

------

xieyuheng


On Tuesday, June 19, 2018 at 2:58:05 AM UTC+8, Matt Oliveri wrote:
>
> Hi, Xieyuheng.
>
> On Monday, June 18, 2018 at 12:32:14 PM UTC-4, xieyuheng wrote:
>>
>> is there a categorical construction to generalize arrow composition,
>> by allowing domain and codomain to be refined (or changed) by the
>> composition ?
>>
>> this construction would be useful for
>> forming theoretical background of dependent type system.
>>
>> for example, compose two functions
>> f : (A x -> B y)
>> g : (B n -> C z)
>> will give us a function of type (A n -> C z)
>>
>
> I can't figure out what rule you're using to get that type. If n unifies 
> with y, how did x become n also? That's what I'm guessing you're trying.
>
>
>> another example would be the following generalized composition in 
>> cartesian closed category :
>>         f   : (t1, t2) -> (t3, t4)
>>         g   : (t, t3, t4) -> (t6, t7)
>>         f;g : (t, t1, t2) -> (t6, t7)
>> and
>>         f   : (t1, t2) -> (t, t3, t4)
>>         g   : (t3, t4) -> (t6, t7)
>>         f;g : (t, t1, t2) -> (t, t6, t7)
>> this can be called `cut`
>> because it looks like gentzen's cut rule in sequent calculus,
>> and it can be used to provide semantic
>> for a stack based concatenative programming language.
>>
>
> This looks more like a combination of composition and tensoring, in a 
> monoidal category. Based on some stuff in a blog post (
> https://golem.ph.utexas.edu/category/2017/11/the_polycategory_of_multivaria.html), 
> the cut rule is instead composition in a polycategory, and you can only cut 
> out one type at a time.
>
> I'm thinking the first "generalized composition" could be explained thus:
> f only looks at the top two stack elements, and replaces them with three 
> others. But the stack generally has additional elements, which I'll 
> abstract away as Γ:
> f  :  (Γ, t1, t2) -> (Γ, t3, t4)
>
> g has a similar story, but we use a different stack/context variable to 
> avoid accidental constraints:
> g  :  (Δ, t, t3, t4) -> (Δ, t6, t7)
>
> To compose these, we need (Γ, t3, t4) to unify with (Δ, t, t3, t4), so (Γ 
> := Δ, t), and
> f;g  :  (Δ, t, t1, t2) -> (Δ, t6, t7)
> which we shorten to (t, t1, t2) -> (t6, t7).
>
> So your composable operators seem to be substitutions (morphisms between 
> contexts) that are polymorphic in most of the context. All but a finite 
> chunk all the way on the right. Composition would presumably be composition 
> of substitutions, after instantiating them with the principal unifier.
>
> For simply-typed systems, contexts and substitutions can be modeled in a 
> monoidal category, which includes cartesian closed categories. For 
> dependent type systems, other kinds of categories are used, like contextual 
> categories and categories with families; hopefully what you want is still 
> instantiation and composition of this new kind of substitution.
>
> I'm not an expert on the kinds of categorical structures involved, but 
> hopefully polymorphic substitutions is a good way of thinking about what 
> you're doing, which connects it with categorical models.
>
> Beware that with dependent types, there generally are not principal 
> unifiers, due to binding operators appearing in types. But concatenative 
> people don't seem to like binders, so maybe you dodged the bullet.
>

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