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,

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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.

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(iii) What significant mathematical examples can you give of your notion ?

See (i).

Best regards,

Richard

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