Re: Terminology of locally small categories without replacement

[Richard Garner <richard.garner@mq.edu.au>]

Dear Jean,

> It is incorrect because Fib(S) does not have pullbacks or equalizers hence
> it not finitely complete

Yes, that's true; however Fib(S) does have PIE limits (or even just
bilimits) which is enough for Ross's definition to make sense. In
fact, one need not even assume the existence of any limits at all: an
object may be defined to be locally small just when, for every cospan
of arrows with small domain, the comma object exists and is again
small.

> It is incomplete for two reasons:
> (i) When I asked for significant mathematical examples, I meant apart from
> locally small fibrations.

In fact there are no other examples; the two notions are essentially
equivalent. Given the 2-category K with a class of small objects
therein, we can consider the full sub-2-category of K spanned by those
objects, and the underlying ordinary category C of this. Now each
object x in K induces a fibration p: E -->  C whose total category E
has as objects, morphisms f: c --> x in K with small domain; and as
arrows (f, c) --> (f',c'), pairs of a morphism h: c --> c' and a
2-cell f => f'h in K. It's now not hard to show that this fibration
will be locally small just when the object x is locally small in the
sense described by Ross (under the additional, and reasonable
assumption that the class of small objects is dense in K---which in
particular is the case when K = Fib(S) and C are the representable
fibrations).

Richard


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