Re: Fibrations in a 2-category

[David Roberts <droberts@maths.adelaide.edu.au>]

I was I think a bit hasty in my last post. I thought it was possible
to separate the use of Choice in the metalogic and in Set, but I can't
see how to stop Choice 'filtering down'. However if we work not with
categories of sets but more general categories, I can get a more
definite answer.


Let S be a site (with a subcanonical singleton pretopology) so that
the bicategory Cat_ana(S) is defined, and also assume that
coequalisers of reflexive pairs exist in S.

Theorem: If Cat_ana(S) is equivalent to a 2-category Cat(C) for some
category C with finite products and coequalisers of reflexive pairs,
then covers are split in S.

Proof:

  In what follows an _equivalence_ of bicategories is defined to be a
2-functor (weak or strict) which is essentially surjective and locally
fully faithful and essentially surjective. If one has Choice in the
metalogic, then one can find a 2-functor which is an inverse up to a
isotransformation etc.

Definition: Let B be a bicategory. An object x in B is called a
_discrete object_ if B(w,x) is equivalent to a set for all objects w.

Let do(B) denote the full sub-bicategory on the discrete objects. For
any object a in a category C there is a discrete object disc(a) in
Cat(C), and disc:C --> Cat(C) is a functor. There is also a  2-functor
Cat(S) --> Cat_ana(S) for a site S (with subcanonical singleton
pretopology), which is the identity on objects. Discrete objects in
Cat(S) are precisely discrete objects in Cat_ana(S).

Lemma: If B = Cat(C) for some category C with finite products and
coequalisers of reflexive pairs of arrows, then disc:C --> do(Cat(C))
is an equivalence.

Lemma: if B = Cat_ana(S) for some site S with coequalisers of
reflexive pairs of arrows, then disc:S --> do(Cat_ana(S)) is an
equivalence.

Lemma: Let F:B --> B' be an 2-functor. Then there is a 2-functor do(B)
--> do(B') (i.e. discrete objects are mapped to discrete objects). If
F is an equivalence then it reflects discrete objects.

Corollary: If F:B --> B' is an equivalence there is an equivalence
do(B) --> do(B') given by restriction of F.

So if we have an equivalence Cat(C) --> Cat_ana(S) and both C and S
satisfy the conditions of the first two lemmas, we have a co-span of
equivalences

C --> do(Cat_ana(S)) <-- S

Thus if one doesn't mind inverting equivalences as defined here, we
have an equivalence S --> C of categories.

Lemma: Given an equivalence of categories S --> C there is an
equivalence Cat(S) --> Cat(C).

Thus we have an equivalence Cat(S) --> Cat_ana(S). But this implies
that the appropriate version of internal Choice holds in S.      #


Going back to Michal's question, this would imply that in the topos S
all regular epimorphisms split, which is of course not always true.

David




On 17 January 2011 09:21, David Roberts <droberts@maths.adelaide.edu.au> wrote:
> Hi Michal,
>
> it is not *always* false. Consider the topoi Set and Set_choice, where
> the first is the category of sets without choice and the second is
> with choice. Then the bicategory of categories, anafunctors and
> transformations in Set is equivalent (assuming choice in the
> metalogic) to the 2-category of categories, functors and natural
> transformations in Set_choice. This is (essentially) shown by Makkai
> in his original anafunctors paper.
>
> However, I doubt that it is always true (only a hunch). Also, one does
> not need a topos as an ambient category in which to define
> anafunctors, only a site where the Grothendieck pretopology is
> subcanonical and singleton (single maps as covering families). The
> topos case is when you take the regular pretopology.
>
> And although you did not ask for a reference, here's one:
>
> http://arxiv.org/abs/1101.2363
>
> which builds on internal anafunctors introduced here
>
> http://arxiv.org/abs/math.CT/0410328
>
> and Makkai's original paper is available in parts from here:
>
> http://www.math.mcgill.ca/makkai/anafun/
>
> David
>

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