Re: Fibrations in a 2-category

[Michael Shulman <mshulman@ucsd.edu>]

Dear Jean,

One way to deal with the difficulty you mention is by using
"anafunctors," which were introduced by Makkai precisely in order to
avoid the use of AC in category theory.

An anafunctor is really a simple thing: a morphism in the bicategory
of fractions obtained from Cat by inverting the functors which are
fully faithful and essentially surjective.  It can be represented by a
span  A <-- F --> B  whose left leg is fully faithful and surjective
on objects.  One intuition is that the objects of F over a\in A are
different "ways to compute a value" of the anafunctor at a.  Different
"ways to compute a value" may give different values, but they will be
canonically isomorphic.

For example, let P --> 2 be a fibration, with fibers B and A.  Then
there is (without AC) an anafunctor A --> B, where the objects of F
are the cartesian arrows of P over the nonidentity arrow of 2, and the
projections assign to such an arrow its domain and codomain.

More generally, if Cat_ana denotes the bicategory of categories and
anafunctors, then from any fibration P --> C we can construct (without
AC) a pseudofunctor C^{op} --> Cat_ana.  Moreover, if we allow
morphisms between fibrations to be anafunctors as well, then the
bicategory of fibrations over C is biequivalent to the bicategory of
pseudofunctors C^{op} --> Cat_ana.  (This should not be read as saying
anything more than it says; in particular I would not claim that
fibrations are always "the same as" indexed categories even from this
viewpoint.  For fixed C, they form equivalent bicategories, which
makes them sufficiently "the same" for some purposes, but, as you have
pointed out, not for other purposes.)


Similarly, regarding "internalization," any ordinary (non-cloven)
fibration does give rise to an internal fibration in the bicategory
Cat_ana.  The same is true for internal fibrations and anafunctors in
a topos (the relevant "non-cartesian" parts of the internal logic of
the topos E having been incorporated into the definition of
Cat_ana(E)).  Unfortunately, since Cat_ana is only a bicategory, not a
strict 2-category, we do not get the strict notion of internal
fibration, but the weaker version as defined by Street, in which
cartesian liftings exist only up to isomorphism.  I think this is a
nice example of when one may be "forced" to use Street fibrations
rather than Grothendieck ones (never claiming, of course, that there
is anything necessarily "wrong" with Grothendieck fibrations when they
suffice).

For example, if p: P --> C is a (Grothendieck) fibration, f: A --> C
and g: A --> P are functors and m: f --> pg is a natural
transformation, then we can define an anafunctor A <-- H --> P in
which the objects of H are pairs (a,n), where a is an object of A and
n: x --> g(a) is a cartesian arrow in P with p(n) = m_a.  The functor
H --> A is surjective on objects because p is a fibration.  Then the
composite anafunctor ph: A --> C is naturally isomorphic to f, and
there is a natural transformation from h to g which lies over m
(modulo this isomorphism) and which is cartesian in Cat_ana(A,P) over
Cat_ana(A,C).  One can generalize to the case when f, g, and p are
also anafunctors and p is a Street fibration (suitably interpreted for
an anafunctor).

In general, it seems to me that there are two overall approaches to
doing category theory without AC (including with internal categories
in a topos):

1) Embrace anafunctors as "the right kind of morphism between
categories" in the absence of AC.  As I mentioned above, many familiar
facts about category theory which normally use AC remain true without
it, if all notions are replaced by their corresponding "ana-"
versions.  Of course, this approach has the disadvantage that
anafunctors are more complicated than ordinary functors, and form a
bicategory rather than a strict 2-category; thus one may be forced
into using other weaker notions like Street fibrations, bilimits, etc.

2) Insist on using only ordinary functors, so that we can work with
the strict 2-category Cat, which is simpler and stricter than Cat_ana.
  However, many theorems which are true under AC now become false.  In
addition to the properties of fibrations as above, one also has to
distinguish between "having limits" in the sense of "every diagram has
a limit" versus the sense of "there is a function assigning a limit to
every diagram."

Personally, while there is nothing intrinsically wrong with (2), I
think (1) gives a more satisfactory theory.  It also has connections
to applications outside of category theory.  For instance, anafunctors
between internal categories in a topos are more or less equivalent to
morphisms between their stack completions, and in various parts of
mathematics internal categories, and notions equivalent to
anafunctors, are frequently used as representatives of stacks (Lie
groupoids, Hopf algebroids, moduli stacks, etc.).  So it is not just a
philosophical reason to prefer (1).  However, I respect that others
may disagree, and I'd be interested in hearing about mathematical
reasons to prefer (2).

Regards,
Mike



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