Locally small categories without replacement, or anything

Locally small categories without replacement, or anything

[JeanBenabou]

Dear Colin,

I gave 36 years ago a definition of locally small fibration over an  
arbitrary base category S. Now look at the following very special  
case: S = Set, C is a category, and  P: Fam(C) --> S the canonical  
fibration where the fiber over the set I is the category C^I. To say  
that this P is locally small in my sense coincides exactly with the  
"more general" notion of C being locally small that you suggest.
Note that my definition of locally small fibration does not suppose  
that S has a terminal object 1, let alone that 1 is a strong  
generator in S. To show that this definition is equivalent to the  
"usual" one, you need not only a replacement scheme in Set, but also  
the fact that 1 is a strong generator in Set.
Thus I think that the correct general definition of "local smallness"  
is the one I gave for fibrations.
As a side important remark, the identity fibration Id(S): S --> S is  
always locally small without any assumption on S, in particular  S  
need not have a terminal object, pull-backs or any kind of limit.  
None of this is true with any of the "variants" of my definition you  
can find e.g. in the Elephant, where you have to assume that S has  
finite limits.
Thus "evil" fibrations can be interesting after all.

Best to all,


Le 1 déc. 10 à 23:00, Colin McLarty a écrit :


> Locally small categories are always defined as categories such that:
>
> LS) for any objects A,B there is a set of all arrows A-->B.
>
> When the base set theory includes the axiom scheme of replacement that
> is equivalent to a prima facie stronger property:
>
> ??) for any set of objects there is a set of all arrows between them.
>
> These two are not equivalent in the absence of the axiom scheme of
> replacement.  There the second is much stronger, but it remains
> important.  Is there a good term for it?
>
> thanks, Colin
>





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