[Agda] Re: Forall and product

[Andreas Abel]

I am using terminology compatible with the rest of IT. ;-)

On 18.02.2015 15:11, Martin Escardo wrote:
> It would be good if people used terminology compatible with the rest of
> mathematics:
>
> https://en.wikipedia.org/wiki/Cartesian_product#Infinite_products
> https://en.wikipedia.org/wiki/Disjoint_union
>
> M.
>
> On 18/02/15 11:20, Martin Escardo wrote:
>>
>>
>> On 18/02/15 11:03, Andreas Abel wrote:
>>> My verbalization is
>>>
>>>    Pi    : "dependent function type"
>>>    Sigma : "dependent pair type"
>>>
>>> which is non-ambiguous.  I also avoid "dependent product".
>>
>> I agree with that.
>>
>> But when you use type theory (and Agda in particular) to do
>> mathematics, you may come across things such as "a product of compact
>> spaces is compact". What is meant in general is that Pi(i:I). C i is
>> compact if each C i is compact, for any I-indexed family C. And in
>> particular that A x B is compact if A and B are compact,
>> regarding A x B as a Pi type as in (3) below.
>>
>> Martin
>>
>>
>>
>>> Cheers,
>>> Andreas
>>>
>>> On 17.02.2015 21:51, Martin Escardo wrote:
>>>>
>>>>
>>>> On 17/02/15 20:15, N. Raghavendra wrote:
>>>>> At 2015-02-17T12:22:58-06:00, Chris Jenkins wrote:
>>>>>
>>>>>> The problem here is in terminology, unfortunately. The "dependent
>>>>>> product" is the pi type (dependent function type), which has a
>>>>>> logical interpretation of "for all". The "dependent sum" is the sigma
>>>>>> type (dependent pair type), which as a logical interpretation of
>>>>>> "there exists". This conflicts with referring to (dependent) pairs as
>>>>>> a (Cartesian) product.
>>>>>
>>>>> I don't know if there's a conflict.
>>>>
>>>> There *is* a conflict in the *use of terminology*: some people use
>>>> "product" to mean Pi, and other people use "product" to mean "Sigma".
>>>>
>>>>> If A and B are two sets, their Cartesian product AxB equals the sum
>>>>> or disjoint-union of the constant family (B(a))_{a:A} which is given
>>>>> by B(a)=B for all a in A.  So the Cartesian product AxB is a sum
>>>>> also, no?
>>>>
>>>> And there is an overlap in the *notions* conveyed by the terminologies,
>>>> as well.
>>>>
>>>> (1) A + B can be represented by a Sigma type indexed by the two-point
>>>>      type 2 with points 0,1:
>>>>
>>>>             C : 2 -> Set
>>>>             C 0 = A
>>>>             C 1 = B
>>>>             A + B = Sigma(n:2).C n.
>>>>
>>>> (2) A x B can be represented by a Sigma type indexed by A, as you say:
>>>>
>>>>             C : A -> Set
>>>>             C a = B
>>>>             A x B = Sigma(a:A). C a.
>>>>
>>>> (3) A x B can also be represented as a Pi type. Take C as in (1):
>>>>
>>>>             C : 2 -> Set
>>>>             C 0 = A
>>>>             C 1 = B
>>>>             A x B = Pi(n:2). C n.
>>>>
>>>> The relevant particular cases, in my view, are (1) and (3). The
>>>> situation (2) can be regarded as a sort of a (useful) "coincidence".
>>>>
>>>> I would advocate calling Pi types "product types" and Sigma types "sum
>>>> types". But then we can't change established terminologies, even if
>>>> they *are* conflicting: some people mean Sigma when they say "product
>>>> types" and other people mean Pi. (And I think it is the people who
>>>> mean Pi are the ones who are terminologically correct, in the sense of
>>>> being terminologically compatible with e.g. category theory.)
>>>>
>>>> The only way out is to avoid the terminology "product" (unless you
>>>> know your audience), at this stage, and instead say "Pi types" and
>>>> "Sigma types".
>>>>
>>>> M.
>>>> _______________________________________________
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>>>> Agda at lists.chalmers.se
>>>> https://lists.chalmers.se/mailman/listinfo/agda
>>>>
>>>
>>>
>>
>


-- 
Andreas Abel  <><      Du bist der geliebte Mensch.

Department of Computer Science and Engineering
Chalmers and Gothenburg University, Sweden

andreas.abel at gu.se
http://www2.tcs.ifi.lmu.de/~abel/