Re: small2

[Marta Bunge <marta.bunge@mcgill.ca>]

This is in reply to a posting by Bob Pare. 

Dear Bob: 
You wrote: 
“As usual it helps put things in relief to generalize them. Consider category theory in a world based on a topos S. A small category is a category object in S. A large category like S or Group(S) is an indexed category, given by a pseudo-functor S^op -> Cat, or a fibration over S if you prefer. A small category C gives a large one by homming. But something is lost in the process.” 
That is precisely how I always thought of small versus large relative to a topos S (the universe of discourse). Further, a large category A (fibered over S) "is" small if and only if the corresponding pseudo-functor A: S^op--> Cat is representable. As you say, a small category C (internal to the topos S) may always be considered as a large one (via its externalization [C]: S^op ->Cat). 
These considerations came up in my and our joint work on stacks (Cahiers, 1979), and more recently in my work with Claudio Hermida on 2-stacks. The notions of a stack and of a 2-stack are taken  to be intrinsic to a topos S, that is, relative to the topology of its (regular) epimorphisms, as introduced by Lawvere 1974. 
Dimension 1. A small category C (internal to a topos S) always has a stack completion when regarded as a large category via its externalization [C]. The stack completion of C is given by yon: [C] -> [C]*= LocRep(S^(C^op)), a weak equivalence functor. Applied to a groupoid G, this gives the classification theorem for G-torsors (Diaconescu 1975). An axiom of stack completions (ASC) in its rough form says that S satisfies it if for ever small category C in S, the fibration LocRep(S^{C^op}) is representable by a category C* so that [C]* and [C*] are equivalent as fibrations. As shown by Joyal and Tierney (1991) by means of Quillen model structures, but also by a general argument involving the existence of a set of generators, (Duskin 1980), Grothendieck toposes satisfy (ASC). 
Dimension 2. The 2-dimensional analogue of the above set-up was discussed in my lecture at CT 2008 (joint work with Claudio Hermida). Our main result is that, for a topos S satisfying (ASC), any 2-category 1-stack C in S, regarded as a 2-fibration, has a 2-stack completion, to wit yon: [C] -> [C]*=LocRep(Stack^(C^op)), a weak 2-equivalence 2-functor. Applied to a 2-gerbe G and suitably interpreted, this gives a classification theorem for G-2-torsors. The validity of an appropriately formulated (ASC)^2 for a Grothendieck topos S (see slides for my lecture at CT 2008) is true by a general argument involving the existence of a set of generators. What is still missing, however, is a construction of a small 2-category 1-stack C*representing the 2-fibration [C]*= LocRep(Stack^(C^op)) in the case of a Grothendieck topos S. The Quillen model structure on 2-Cat given by Lack 2002 is not suitable for this purpose. 

Dimension n. Analogue results in higher dimensions are less tractable but a pattern emerges from the passage from dimension 1 to dimension 2.  

Remark (concerning a previous posting of yours): Although using the S-indexed versions of fibrations over S is useful, just as presentations of groups are useful, the entire discussion of stacks can be carried out at the level of fibrations (ditto 2-fibrations). I fail to understand what is all the fuss about the use of S-indexed categories if taken in that spirit. Certainly not deserving ridicule!

 You also wrote: 
“Well this is my second posting in a week bringing my life-time total to three! Seems a good place to stop.”

 To me, that is a poor reason to give. This forum could certainly profit from your interventions, but I understand your qualms, as I myself quite often abstain from an urge to intervene. 

 Happy New Year to you and everyone, Marta


************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
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Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
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