Re: Small is beautiful

[burroni@math.jussieu.fr]

Dear categorists,

The question for me is not : small or large categories, but small or  
large structures.

The Bourbaki's time was the time of small structures (groups, monoïds,  
rings, spaces, etc), the "categorical time" is the time of structures  
of structures,i.e. large structures (groupoids, complete categories,  
abelian catégories, topos, etc.).
The bridge beetwen the firsts and the seconds is the Yoneda lemma,  
that is to say the introduction of logic, thus of the only true evil  
category : the category of sets. All the other large categorical  
stuctures introduced by categorists are deduced from category Set, by  
abstraction, generalisations, constructions or restrictions. In fact  
"category theory" is an (wonderfull but) inappropriate name (here the  
word category is only an important and historical keyword): the reason  
is that a lot of data (limits, classifiant object, etc) appear as  
properties because of their universal properties (unicity up to  
isomorphism). But "category theory" is an illusion, nobody studies  
seriouly the categorical structures (I don't know any structure  
theorem on the categorical structures without additional properties).
That is my starting point of  reflexion on this subject.

I think there is a true theory of the small categories, but it is not  
yet born, if it must ever exist. This theory should be, not only for  
the 1-categories, but also for the n and omega-categories (strict or  
not possibly). And, for me, the Yoneda lemma is an important tool (but  
only a tool). Such a theory may be eventually important for the  
computer science (particularly for the formal languages).

My best wishes for the new year,

Albert burroni


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