Re: Terminology of locally small categories without replacement
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean Thank you for your message. I have been in Canberra and am only now catching up with emails, reference writing and the like. In the meantime, I see that you provided an answer to the original question under this "Subject" (involving locally small fibrations) and that Richard Garner has responded well to your questions showing the close relationship between the concepts. I think the comma object observation (I would not claim it competitively as "my notion" particularly) was a helpful viewpoint for some researchers. As Richard points out that, to make the definition, the 2-category K does not really need any limits however comma objects are useful (in the same way that internal category can be defined in any category but having pullbacks makes one feel better). (I now notice in my message that C somehow was mistyped later as R. As indicated, more conditions on C give stronger consequences.) I would be pleased to hear what you have in mind as some of the significant and numerous results derivable using locally small fibrations. Walters and I were interested at one time in developing category theory in a 2-category with an analogue P of the presheaf construction ("Yoneda structures"). Mark Weber has recently been able to make use of some of these ideas in developing foundations for recent advances in Batanin's operad theory. I believe my paper [The petit topos of globular sets, J. Pure Appl. Algebra 154 (2000) 299-315] was some help in this respect. In any case, one example of a 2-category K is provided by a finitely complete cartesian closed category E with an internal full subcategory S; we take K = Cat(E) and Pa = [a opposite,S] where [ , ] is cartesian internal hom in Cat(E). In particular, we can take E = Cat so that K is the 2-category Dbl of double categories. Given any internal full subcategory set of Set, there is an internal subcategory fun of Cat which is the double category of squares based on categories in set. The objects of fun are categories in set, the horizontal and vertical morphisms are functors, and the squares are natural transformations in the squares. The small objects of K = Dbl are defined to be the double categories in set. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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Richard Garner -
Ross Street