Re: Terminology of locally small categories without replacement

[JeanBenabou <jean.benabou@wanadoo.fr>]

Dear Richard,

Thank you for your quick answer, which unfortunately is both  
incorrect and incomplete.

It is incorrect because Fib(S) does not have pullbacks or equalizers  
hence it not finitely complete
It is incomplete for two reasons:
(i) When I asked for significant mathematical examples, I meant apart  
from locally small fibrations, because I do not believe in abstract  
nonsense "generalizations" which have no genuine examples except well  
known special cases.
(ii) In my mail there was another question which you seem to have  
forgotten namely: What can you prove about the locally small objects  
of K, especially since you assume nothing on C ?
To complete my remark (ii) I would mind a little less the lack of  
genuine examples of this generalized notion if at least under the  
mere assumptions of Street on could prove a few non totally trivial  
results.
I would like to point out for example that, with Street's definition,  
one cannot even prove that a small object of K is locally small.

I'm sure that Ross, who gave this definition, will very soon give a  
correct and complete answer to the three questions I asked him in my  
previous mail.

Best regards,

Jean


Le 7 déc. 10 à 23:22, Richard Garner a écrit :

> Dear Jean,
>
>> Could you please tell me:
>> (i) Given a category S how does one chose the finitely complete 2-
>> category K and the class C of small objects so that the locally small
>> objects of K in your sense, are the locally small fibrations over S ?
>
> Take K = Fib(S) and take C to be the representable fibrations. For me
> this is actually the easiest way to remember the definition of locally
> small fibration.
>
>> (iii) What significant mathematical examples can you give of your
>> notion ?
>
> See (i).
>
> Best regards,
>
> Richard



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