Cat as a '2-fibration' over Set

Cat as a '2-fibration' over Set

[David Roberts]

Hi all,

To start with think of Cat as a 1-category. The functor Obj:Cat \to
Set sending a small category to its set of objects is a fibration.
This can be easily seen by constructing, given a category C = (C_1
\rightrightarrows C_0) and a function f:A \to C_0, the set of arrows
A^2 \times_{f,C_0^2}C_1 (the pullback of (s,t):C_1 \to C_0^2) of the
category C[f].

The cartesian lift of f is then the canonical functor F:C[f]\to C.

Now given another function g:A\to C_0 -- giving rise to G:C[g]\to C --
and a natural transformation F \Rightarrow G there is a canonical
isomorphism C[f]\simeq C[g] over C. Thus if we think of Cat as a
2-category, there is something extra going on. For example, one gets a
pseudofunctor Set \to 2Cat on choosing specified pullbacks to define
C[f].

Has this phenomenon been studied before? (I would think so)
Does this make Obj a fibration of 2-categories (see e.g. Hermida, or Bakovic)?
Or is this a more 'classical' concept? More basically, where was this
fact first pointed out?

David


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Re: Cat as a '2-fibration' over Set

[Ross Street]

On 07/10/2010, at 8:18 PM, David Roberts wrote:

> To start with think of Cat as a 1-category. The functor Obj:Cat \to
> Set sending a small category to its set of objects is a fibration.


Dear David

In a daring version of an undergraduate algebra unit on groups,
I taught the notions of cartesian and opcartesian morphism
for a functor and looked at them for the functor ob : Cat --> Set.
The goal was to give a groupoid proof of the Nielsen-Schreier
theorem using fibrations in the small (between groupoids) and in the
large. I achieved the goal to my own satisfaction; I think most of
the students thought otherwise. A core of them liked it. This is
the most explicit category theory I have tried to teach pre fourth
year honours.

My inspiration very definitely came from Ronnie Brown's topology  
book(s).

I'm not at work today (Saturday, and a grandson's birthday party)
so I can't check whether these constructions of direct and inverse
images for ob : Cat --> Set are in that book, whether it is the
ob : Gpd --> Set  case that is there, or what. Ronnie can tell us
perhaps. Anyway, it is essentially there. It may not be phrased in
terms of cartesian morphisms.

> Has this phenomenon been studied before? (I would think so)
> Does this make Obj a fibration of 2-categories (see e.g. Hermida, or  
> Bakovic)?
> Or is this a more 'classical' concept? More basically, where was this
> fact first pointed out?

I too would like to know of other references.

I am ashamed to say I hadn't thought about the 2-fibrational aspects
of ob : Cat --> Set.

Also, how about the Beck-Bénabou-Roubaud-Chevalley condition?

Ross



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Re: Cat as a '2-fibration' over Set

[Ronnie]

Dear All,

There is a strong emphasis on cocartesian morphisms of groupoids  in the 
various editions of `Topology and Groupoids' (but called there 
`universal morphisms'), following Philip Higgins' paper  1964 paper on 
groupoids. Philip's work on this idea was nicely jacked up by him to 
induced morphisms of crossed modules, i.e. using the cofibration XMod 
\to Group(oid)s giving the base group (or groupoid!), with a good 
application to 2nd relative homotopy groups, and this appeared in our 
papers. In higher dimensions, you get the Relative Hurewicz Theorem this 
way.

One of the applications to groupoids I like is that the cocartesian 
morphism from the groupoid I (indiscrete on 0,1) over the identification
{0,1} \to {0} gives I \to Z= integers. This seems a enough good reason 
why the fundamental group of the circle is the integers!

Doing this for categories instead of groupoids gives of course `2' \to N 
is cocartesian over the same identification, which thus gives another 
formulation of induction!

In the new book `Nonabelian algebraic topology' (in final stages, 
downloadable from my web page, and final comments welcome) we emphasise 
the fibrations and cofibrations of categories approach.

Ross asks about the Beck-Bénabou-Roubaud-Chevalley condition: I would 
like to know of applications to the matters considered in these two books!

Ronnie




On 09/10/2010 07:12, Ross Street wrote:
> On 07/10/2010, at 8:18 PM, David Roberts wrote:
>
>> To start with think of Cat as a 1-category. The functor Obj:Cat \to
>> Set sending a small category to its set of objects is a fibration.
>
>
> Dear David
>
> In a daring version of an undergraduate algebra unit on groups,
> I taught the notions of cartesian and opcartesian morphism
> for a functor and looked at them for the functor ob : Cat --> Set.
> The goal was to give a groupoid proof of the Nielsen-Schreier
> theorem using fibrations in the small (between groupoids) and in the
> large. I achieved the goal to my own satisfaction; I think most of
> the students thought otherwise. A core of them liked it. This is
> the most explicit category theory I have tried to teach pre fourth
> year honours.
>
> My inspiration very definitely came from Ronnie Brown's topology book(s).
>
> I'm not at work today (Saturday, and a grandson's birthday party)
> so I can't check whether these constructions of direct and inverse
> images for ob : Cat --> Set are in that book, whether it is the
> ob : Gpd --> Set  case that is there, or what. Ronnie can tell us
> perhaps. Anyway, it is essentially there. It may not be phrased in
> terms of cartesian morphisms.
>
>> Has this phenomenon been studied before? (I would think so)
>> Does this make Obj a fibration of 2-categories (see e.g. Hermida, or 
>> Bakovic)?
>> Or is this a more 'classical' concept? More basically, where was this
>> fact first pointed out?
>
> I too would like to know of other references.
>
> I am ashamed to say I hadn't thought about the 2-fibrational aspects
> of ob : Cat --> Set.
>
> Also, how about the Beck-Bénabou-Roubaud-Chevalley condition?
>
> Ross
>
>


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