Re: Fibrations in a 2-category

[Michael Shulman <mshulman@ucsd.edu>]

On Mon, Jan 17, 2011 at 1:02 AM, David Roberts
<droberts@maths.adelaide.edu.au> wrote:
> 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.

I don't think this argument quite works, but I think one can show
something almost as good, namely that if S is a topos, C has finite
limits, and Cat_ana(S) is equivalent to Cat(C), then S satisfies the
*internal* axiom of choice.

> 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.

This isn't quite right; a discrete object in Cat(C) is essentially an
internal equivalence relation in C, and so do(Cat(C)) is the category
of equivalence relations and functors between them (not morphisms
between their quotients).  This is a full subcategory of the free
exact completion C_{ex/lex}, which contains the free regular
completion C_{reg/lex} (the category of kernels).

We can say that disc:C --> do(Cat(C)) is fully faithful, and its
essential image consists of the projective objects in do(Cat(C))
(assuming that C has finite limits).  In particular, do(Cat(C)) has
enough projectives.

> 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.

This I believe if S is exact, such as a topos.  The point is that an
effective equivalence relation in  S, regarded as an internal category
in S, admits a surjective weak equivalence to its quotient object,
regarded as a discrete internal category.  Thus, the two become
equivalent in Cat_ana(S), though not in general in Cat(S).

> 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.

An equivalence of bicategories certainly preserves and reflects
discrete objects, which is all that matters for this proof.  (But a
general functor of bicategories need not preserve discrete objects.)

> So if we have an equivalence Cat(C) --> Cat_ana(S)

... then we can conclude that S, being equivalent to do(Cat_ana(S))
and hence to do(Cat(C)), has enough projectives, namely C.

Already this is a nontrivial restriction on a topos (or set-theoretic
axiom), although it can hold in the absence of IAC.  However, we can
say more.

First, if we identify C with the subcategory of projectives in S, then
the equivalence functor Cat(C) --> Cat_ana(S) must be, up to
equivalence, the inclusion which regards internal categories in C as
internal categories in S, and internal functors as internal
anafunctors.  For being an equivalence, it in particular preserves lax
codescent objects; but every internal category is a lax codescent
object formed of discrete internal categories, and the functor C -->
Cat(C) --> Cat_ana(S) is what we used to identify C with the
projective objects of S.

Therefore, since this functor is an equivalence, every internal
category in S must be equivalent, in Cat_ana(S), to an internal
category in C, i.e. an internal category in S formed of projective
objects.  Now for any object A of S, we have an internal category
1+A+1 \rightrightarrows 1+1 with "two objects" and A as the
object-of-morphisms from one to the other (and only identity arrows
otherwise).  If this category is equivalent in Cat_ana(S) to one
composed of projective objects, then we must have a surjective weak
equivalence to it from such a category, which is equivalent to giving
a well-supported projective object P such that PxPxA is projective.
Thus, any object A is "locally projective", which is sufficient for
IAC.

(If we are talking about set theoretical foundations, rather than
working in a topos, we could then pick an element p of P, which exists
since it is well-supported.  Then since the projectives are closed
under finite limits, the fiber of PxPxA over (p,p), namely A, would be
projective, and hence AC holds.)


I also think it's worth mentioning that if S merely has enough
projectives, then we can identify Cat_ana(S) (up to equivalence of
bicategories) with a full sub-2-category of Cat(S), consisting of
those internal categories whose object-of-objects is projective (but
with no condition on the object-of-morphisms).

In fact, such categories are the cofibrant objects in a model
structure on Cat(S), in which everything is fibrant and whose weak
equivalences are the internally fully-faithful and
essentially-surjective functors.  Thus, this is a particular case of
the fact that morphisms in the homotopy (2-)category of a model
category are represented by maps from a cofibrant replacement to a
fibrant replacement.

(When S is a Grothendieck topos, there is also a model structure on
Cat(S) with those weak equivalences in which every object is
cofibrant, and in which the fibrant objects are stacks.  I believe
this was proven by Joyal and Tierney in their paper "Strong stacks and
classifying spaces".)

The set-theoretic axiom that "there exist enough projective sets" is a
weak form of choice called the "presentation axiom" or "COSHEP" ("the
Category Of Sets Has Enough Projectives").  It implies dependent
choice and some other weak forms of choice, and tends to hold in
models arising from type theory.  So if one is willing to accept that
axiom in lieu of full AC, or one is working in a topos that has enough
projectives (such as, notably, the effective topos), then one can
avoid talking about anafunctors by restricting to internal categories
with projective object-of-objects.  I don't know whether there is a
dual set-theoretic "axiom of small stack completions".

Regards,
Mike


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