Re: small2

Re: small2

[Marta Bunge]

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


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Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
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Montreal, QC, Canada H3A 2K6
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Small2

[Robert Pare]

Thanks to all who replied to my posting either privately or on Categories.

I'd like to clarify a few points. Ross's interpretation of what I meant,
and which Vaughan agrees with, is not what I was trying to say. It is part
of the picture but not the one I was promoting.

Small categories play a different role in category theory than large
categories. Small categories are used for indexing things. Large
categories consist of the things we are indexing. Okay, syntactic was
not the right word. Combinatorial might be better although that has
finiteness overtones. Strict? Well, I'll just stick with small. Small
is not the same as essentially small. As Jeff Egger pointed out, the
category of finite sets is not small. It's not even clear what this category
is. The ZFCists would say that for each set A we get a finite set {A} and
another {{A}}, and Barwise might even wonder if these last two are distinct.
But I'm digressing.

Of course every small category can be considered as a large one (perhaps
large isn't the right word either). Then two equivalent ones would be
considered "the same". I don't think that this is how the working
categorician works. Sometimes equivalent categories are "the same" and
sometimes not. An equivalence relation is a category but we would lose
something (everything?) if we identified it with equality in the quotient
set. So I'm saying that, as a matter of "categorical hygiene", we could
be a bit more explicit about what kind "equivalence" we are allowing.
And without changing mathematical practice, I'm advocating that "small"
should imply that it's okay to talk of equality of objects.

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.

The object functor  Cat -> Set is not a 2-functor so if you compose it
with a pseudo-functor S^op -> Cat you get an assignment that doesn't
preserve composition in any sense. So a general indexed category doesn't
have a discrete category of objects. What you can do is take the groupoid
of isomorphisms 2-functor Cat -> Gpd, so that a large has a groupoid of
objects, thinking of a groupoid as a set with a 2-equality on it. An
indexed category gotten from a small category gives an actual functor
S^op -> Cat, so now you can compose with Ob : Cat -> Set so that a
small category has a discrete groupoid of objects.

And that's where the idea of considering categories with a specified
"equality groupoid" of isomorphisms, and equivalences defined to have
the isomorphisms from the corresponding groupoids. Small categories
have only identities, so equivalence means isomorphism, and large
categories have all isos. At the time there were no good examples
of intermediate groupoids (just products of small and large categories
and the like) but Hilbert spaces with unitary isos is a good one.

Well this is my second posting in a week bringing my life-time total
to three! Seems a good place to stop.

Best wishes for the new year. May your happiness be large and your
disappointments small.

Bob



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