Re: Fibrations in a 2-Category

[Michael Shulman <mshulman@ucsd.edu>]

Dear Jean,

As you surmise, anafunctors are more or less the same as representable
distributors.  I apologize for not pointing that out to begin with,
and I'm glad you brought it up.  Specifically, the bicategory Cat_ana
is equivalent to the bicategory Rep of categories and representable
distributors, in the way that you sketched.  So if by

> 3) Work with distributors.

you mean to (among other things) regard Rep as a replacement for Cat,
then I don't really view that approach as very different from my (1).
In particular, it still resolves the issues you raised about internal
fibrations: ordinary fibrations should equally well give rise to
internal (Street) fibrations, in the representable sense, in Rep.

I have no objection to using the language of representable
distributors instead of anafunctors.  And as you point out,
distributors have many other uses -- you don't need to convince me
that distributors are useful and important!  (I feel obliged to point
out that anafunctors can be enriched just as well as distributors can,
but distributors can certainly be used for many things that
anafunctors cannot.)  My experience in mathematics (which is
admittedly much shorter than yours) is that usually, when a given
concept has two equivalent representations, it is useful to know about
both of them and how to go back and forth between them, since each
will have its own advantages and be preferred by different people.

Thus, my answer to your questions "which do you prefer?" is that I
would prefer to have both at my disposal and be able to use either
one.  Without prejudicing anafunctors over representable distributors,
I could rephrase the point I intended to make in my previous email as:
by replacing Cat with a bicategory whose morphisms are "functors
defined up to isomorphism," we can recover many facts about category
theory which classically require AC.  Whether that bicategory is
Cat_ana or Rep is immaterial to the question of "how to do category
theory without AC."


Having said that, I suppose I should nevertheless also say a little
bit about why I like to have anafunctors at my disposal *in addition
to* distributors.  One reason is that sometimes it requires a little
contortion to put something in the form of a representable
distributor.  For instance, if a category A has binary products, then
there is obviously a product-assigning representable distributor P: A
x A -/-> A, defined by

P(a,(a_1,a_2)) = Hom_A(a,a_1) x Hom_A(a,a_2)

But if A has binary coproducts, then in order to define a
coproduct-assigning representable distributor C: A x A -/-> A, then
(as far as I know) one needs to say something like

C(a,(a_1,a_2)) = the set of triples (a_3,p_1,p_2,f) where p_i: a_i -->
a_3 are the injections into a coproduct and f: a --> a_3, modulo an
equivalence relation (a_3,p_1,p_2,f) ~ (a_3',p_1',p_2',f') if there
exists a (necessarily unique iso)morphism g: a_3 --> a_3' commuting
with all the structure maps.

Of course, C is more easily defined as a corepresentable distributor.
But if you want to define a functor that involves both limits and
colimits, like (a,b,c) |--> a x (b + c), then it is not "naturally"
represented as either a representable or a corepresentable one --
although it always *can* be so represented, essentially by passing
across the equivalence Cat_ana = Rep that you sketched.

By contrast, with anafunctors, all of these functors can be
represented "naturally" in analogous ways.  In the first case, we
consider the span AxA <-- P --> A, where P is the category of binary
product diagrams in A.  In the second case, we consider the span AxA
<-- C --> A, where C is the category of binary coproduct diagrams.
And in the third case, we consider the span AxAxA <-- D --> A, where D
is the category of binary coproduct diagrams together with a product
diagram one of whose factors is the vertex of the coproduct diagram.
Note that in each case, the middle category is the category of models
of a "sketch" in A.

Personally, I find the anafunctor way of representing such "functors"
a bit cleaner, and sometimes easier to work with.  Roughly speaking, I
would say that anafunctors are formulated exactly in order to describe
"functors defined up to isomorphism."  Representable distributors, by
contrast, can be described as "functors valued in representable
presheaves."  "Objects defined up to isomorphism" and "representable
presheaves" are *formally* equivalent (without invoking AC), but not
every "naturally occurring" object-defined-up-to-isomorphism is "given
in nature" by the presheaf it represents.  Some are given by the
copresheaf they corepresent; others aren't given directly in either of
those ways.  (But you can guess from the fact that there are lots of
quotation marks in this paragraph, none of this is particularly
formal.  In particular, I don't present it as an argument intended to
convince you to prefer anafunctors over representable distributors,
but rather as a reason why I or someone else might like to think about
anafunctors *in addition to* representable distributors.  If you like,
you can think of an anafunctor as a particularly convenient
"presentation" of a representable distributor.)


I also find it illuminating that amongst all the "classical" facts
about category theory that naively become false without AC, if you
take the single statement "a fully faithful and essentially surjective
functor is an equivalence" and "force" it to be true in a universal
way, then you end up with a world in which all (or at least most) of
the *other* "classical" facts *also* become true again.  As far as I
know, that fact is easiest to express using anafunctors and calculus
of fractions -- although I would be very interested to see a direct
proof that Rep is the result of formally inverting the weak
equivalences in Cat (i.e. a proof that doesn't essentially go by
proving that Rep is equivalent to Cat_ana).


In regards to your other questions:

> Ordinary categories of fractions are
> very complicated, unless you have a calculus of right (or left) fractions.
> Is there, precisely defined, and without neglecting the coherence of
> canonical isomorphisms, such a "calculus" defined. Does it apply to the
> "simple thing" of anafunctors.

I wouldn't use the phrase "category of fractions" unless there is a
calculus of fractions; otherwise I would probably say "homotopy
category" or "localization".  Calculi of fractions for bicategories,
and the construction of anafunctors thereby, can be found in:

Pronk, D.  "Etendues and stacks as bicategories of fractions",
Compositio Math. 1996

Roberts, D.  "Internal categories, anafunctors and localisations",
arXiv:1101.2363

> what is the category of anafunctors with domain the terminal category 1 and codomain a category C?

It's sometimes called the category of "ana-objects" or "cliques" in C.
  Its objects are diagrams in C whose domain is a contractible
category.  In the setting of internal categories in a topos, it is a
stackification of C.  (Does that count as a precise mathematical
application?)

Best,
Mike


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