[Agda] Re: Forall and product

[Martin Escardo]

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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>>>
>>
>>
>

-- 
Martin Escardo
http://www.cs.bham.ac.uk/~mhe