the dual category

[Alexander Kurz <axhkrz@gmail.com>]

I would like to add another example to Eduardo’s.

In computer science both algebras and coalgebras for an endofunctor on sets are useful structures and both initial algebras and final coalgebras play an important role in the semantics of programming languages.

It is now an important feature that algebras and coalgebras over set are not dual to each other. Only the invention of the dual category reveals the underlying duality.

The ensuing tension between `abstract’ duality and `concrete’ non-duality is certainly one reason why the study of set-coalgebras is fascinating. 

For example, whereas it is well-known that the initial sequence of a finitary set-endofunctor converges in omega steps, a result by Worrell shows that the final sequence of a finitary set-endofunctor converges in omega+omega steps.

Best wishes, Alexander

> On 12 Sep 2017, at 17:09, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
> 
> On 11/09/17 13:19, Joyal, Andr? wrote:
>> Dear John,  and category theorists,
>> 
>> The fact that every category has an opposite introduces
>> a symmetry in mathematics that would not be there otherwise.
>>
>> The category of sets is not self dual, but a disjoint union of sets
>> is a coproduct, dual to a product.
>> 
>> Thurston does not show esteem for logic.
>> Most mathematicians are taking logic for granted; they just use it
>> as a part of their natural language.
>> It is obvious that human understanding depends on the
>> the laws of thought, on logic.
>> In a sense, category theory is a branch of mathematical logic,
>> since it greatly improves mathematical thinking in general.
>> A category theorist might say (not too loudly) that mathematical logic
>> is a branch of category theory.
>> 
>> Best,
>> andr?
>> 
> 
> The opposite category (*) may look a senseless obscurity and make some
> people nauseous, but it seems to me it made an important contribution to
> the understanding of mathematics. It took a long time to form part of
> mathematical thinking (and still is). For example, Bourbaki treatment of
> limits (of sets say) define and develops basic properties of projective
> limits, including the universal property. Later does the same for
> inductive limits, and includes a proof of the dual statements !!. He had
> to do so since it had not incorporated categories and the opposite
> category. He states what a universal property is, but can not state that
> the respective universal properties (for limits and colimits) are one
> the dual of the other.
> 
> (*) The axioms of a category are self dual. Another examples are abelian
> categories, and a very subtle one, namely, Quillen's model categories.
> 
> Many categories are not self dual, and this is underneath the duality
> between algebra and geometry.
> 
> best   e.d.
> 

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