[Haskell-cafe] Re: Basic question concerning the category Hask (was: concerning data constructors)

[Jonathan Cast]

On 6 Jan 2008, at 12:27 PM, Dominic Steinitz wrote:

> Jonathan Cast <jonathanccast <at> fastmail.fm> writes:
>
>>
>>>
>>>>> Extensionality is a key part of the definition of all of these
>>>>> constructions.  The categorical rules are designed to require, in
>>>>> concrete categories, that the range of the two injections into a
>>>>> coproduct form a partition of the coproduct, the surjective  
>>>>> pairing
>>>>> law (fst x, snd x) = x holds, and the eta reduction law (\ x ->  
>>>>> f x)
>>>>> = f holds.  Haskell flaunts all three; while some categories have
>>>>> few
>>>>> enough morphisms to get away with this (at least some times),
>>>>> Hask is
>>>>> not one of them.
>>>
>>> That interpretation is not something that is essential in the notion
>>> of category, only in certain specific examples of categories
>>> that you know.
>>
>> I understand category theory.  I also know that the definitions used
>> are chosen to get Set `right', which means extensionality in that
>> case, and are then extended uniformly across all categories.  This
>> has the effect of requiring similar constructions to those in Set in
>> other concrete categories.
>>
>
> Referring to my copy of "Sheaves in Geometry and Logic", Moerdijk  
> and Mac Lane
> state that "in 1963 Lawvere embarked on the daring project of a purely
> categorical foundation of all mathematics". Did he fail? I'm probably
> misunderstanding what you are saying here but I didn't think you  
> needed sets to
> define categories;

Right.  But category theory is nevertheless `backward compatible'  
with set theory, in the sense that the category theoretic  
constructions in a category satisfying ZFC will be the same  
constructions we are familiar with already.  The category-theoretic  
definitions, when specialized to Set, are precise (up to natural  
isomorphism) definitions of the pre-existing concepts of cartesian  
products, functions, etc. in Set.  Or, to put it another way, the  
category-theoretic definitions are generalizations of those pre- 
existing concepts to other categories.  Hask has a structure that is  
Set-like enough that these concepts generalize very little when  
moving to Hask.

> in fact Set is a topos which has far more structure than a
> category. Can you be clearer what you mean by extensionality in  
> this context?

By `extensionality' I mean the equalities which follow from using  
standard set-theoretic definitions for functions, products,  
coproducts, etc. --- surjective pairing, eta-contraction, etc.  My  
understanding is that, in fact, the category-theoretic definitions  
are designed to capture those equations in diagrams that can be used  
as definitions in arbitrary categories.  It's possible to view those  
definitions, then, as more fundamental descriptions of the concepts  
than what they generalize, but the fact that they are generalizations  
of the ideas from Set shows up in categories similar to Set (and Hask  
is certainly more similar to Set than, say, Vec).

jcc