wlawvere | 25 Jul 01:27 2014

Stephen H. Schanuel

Stephen H. Schanuel

Dear colleagues,

It is with deep sadness that I report that my best friend,
Steve Schanuel, died today.

His intellectual generosity and quick mathematical
wit,  legendary among colleagues and students alike,
will continue to inspire many.

Bill Lawvere

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Adam Gal | 24 Jul 16:53 2014
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Preprint: Symmetric self-adjoint Hopf categories and a categorical Heisenberg double

Dear all,

The following preprint is available on the arxiv. We would be really
thankful for any comments or suggestions :)

Symmeric self-adjoint Hopf categories and a categorical Heisenberg double

Adam Gal, Elena Gal

We define what we call a symmetric self-adjoint Hopf structure on a
semisimple abelian category, which is an analog of Zelevinsky's
positive self-adjoint Hopf algebra structure for categories. As
examples we exhibit this structure on the categories of polynomial
functors and equivariant polynomial functors and obtain a categorical
manifestation of Zelevinsky's decomposition theorem involving them. It
follows from the work of Zelevinsky that every positive self-adjoint
Hopf algebra A admits a Fock space action of the Heisenberg double
(A,A). We show that the notion of symmetric self-adjoint Hopf category
leads naturally to the definition of a categorical analog of such an
action and that every symmetric self-adjoint Hopf category admits such
an action

http://arxiv.org/abs/1406.3973

Best regards,
Adam Gal

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(Continue reading)

Adam Gal | 24 Jul 16:52 2014
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Final objects in 2-categories

Hi all,

Has someone studied the notion of final objects in a 2-category? I
know that we can define it using the classifying space, and that in
this sense Quillen's theorem A holds and tells us this is equivalent
to some fibers being contractible.
This seems to be a bit too coarse though. For instance in our recent
paper (with E. Gal) we wanted to prove that something is final, and
what we showed is that these fibers have an initial and final object.
So they definitely have contractible classifying spaces, but it seems
that we can say something more precise than this.

The question is if this fits into some finer notion of final object in
2-categories which has been studied.

Thanks,
Adam

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Jean Bénabou | 24 Jul 05:48 2014
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Re: A remark about Street fibrations

Dear Ross,

I agree with you on all points: 
Pseudo-fibrations is much better than weak fibrations, it describes more precisely the notion defined. I
hope it will be adopted by the category-commmunity.
I am also convinced that these pseudo fibrations have a mathematical role.

The example I gave has two interests:
(i) To show that calling them fibrations would violate all our intuitions about the idea of fibration. 
(ii) to provide a new, important,  and perhaps not known example of such pseudo fibrations.

Of course what makes things work in that example is that, if G is a groupoid, for cospans with codomain G comma
objects and pseudo pullbacks coincide.
This is also true in any 2-Category C, if you define a groupoid of C to be an object G such that for each object X
of C, the category  C(X,G) is a groupoid.

Best wishes,

Jean

Le 24 juil. 2014 à 00:03, Ross Street a écrit :

> On 23 Jul 2014, at 3:42 pm, Jean Bénabou <jean.benabou <at> wanadoo.fr> wrote:
> 
>> In view of this example I suggest that the name of fibrations should be used exclusively for Grothendieck
fibrations, the usual ones or their internalizations along the lines I described, and another name, e.g.
weak fibrations, be given to the notion defined by Street.
>> Woud you agree with this, Ross ?
> 
> Dear Jean
(Continue reading)

Jean Bénabou | 23 Jul 07:42 2014
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A remark about Street fibrations

[Note from moderator: Prof. Benabou's recent posts have been forwarded as 
received by the list. Everyone is reminded of the list policy at: 
http://www.mta.ca/~cat-dist/ ]

In order to stay simple I shall consider only the case where the 2-category with comma objects and
2-pullbacks is Cat, and I shall even accept the axiom of universes and AC.

In that case, but only in that case, the Grothendieck construction makes sense and the theorem, proved by
Grothendieck himself, stating that indexed categories and fibered ones are 2-equivalent is, of course, correct.
Nevertheless, as Dubuc points out, Grothendieck discarded indexed categories for fibrations, and
neither he nor his school ever used indexed categories.
For ages, I have tried to convince people, in particular on this mailing list, that indexed categories
ought to  be abandoned, not only because of the authority of Grothendieck, but for mathematical reasons
which I explained, and I wrote the paper in the Journal of Symblic Logic  (JSL) to explain my views.
This had no effect, and sometimes got me a lot of abuse. In particular the JSL paper was qualified as a
pamphlet, and I still consider it as the deepest paper I ever wrote. That is why I dedicated it to Grothendieck.
I'm glad to see that in the discussion, only fibrations were considered, and such notions  as fibrations
with internal sums and products, which I introduced, were used.

So far, I have had no reaction from Bill Lawvere who introduced indexed categories, and whose influence
determined the wrong choice of indexed versus fibered made by many people.
More surprisingly, no reaction either from Peter Johnstone who made in the Elephant a complete mess of the
whole question. I hope he will give his opinion.

I would like to thank Ross Street for his answer, and make a few comments on his notion of non evil fibration to
answer a question asked by Steve Vickers which I quote:

But that seems to claim that in Cat it doesn't matter whether you use iso or  equality in the Chevalley
condition. Does that accord with your understanding?

(Continue reading)

Ross Street | 22 Jul 23:52 2014
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Re: Composition of Fibrations

Dear Steve and Jean

On 23 Jul 2014, at 12:55 am, Steve Vickers <s.j.vickers <at> cs.bham.ac.uk> wrote:

> In Cat every 0-fibration is normal,

This statement in SLNM420 is false (since in that paper 0-fibration 
means existence of a pseudo-L-algebra structure). 
I must have temporarily believed it when I wrote it.
I was using ``normal’’ to mean strict unit condition.
Grothendieck fibrations with chosen cleavage 
are the normal pseudo-L-algebras.

I apologize profusely for confusion caused.

Ross

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Steve Vickers | 22 Jul 16:55 2014
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Re: Composition of Fibrations

[Ross: Any comments?]

Dear Jean,

I wasn't properly aware of those issues around choosing iso or equality, so it's lucky I got into this
discussion. My intuition of what you are saying is  that with iso, roughly speaking, the reindexing is only
pseudofunctorial. For example, if you pull back along the diagonal B -> ΦB, to get the reindexing along
identities, then you get an endofunctor of each fibre that is isomorphic to the identity. Am I on the right
lines? Is this all written down  somewhere?

After your messages I noticed that Street has a remark after his proposition, whose significance I overlooked:

"Compare the above proposition with Gray [2] p.56; so we have related the definition of 0-fibration here
with the definition of opfibration in [2] when K  = Cat. Notice that the unit of the adjunction l -| p~ for Gray
is not just an isomorphism but an identity. It is worth pointing out the reason for this since we will need
the observation in the next paper. A 0-fibration will be called normal when there is a normalized pseudo
L-algebra structure on it.  In Cat every 0-fibration is normal, but in other categories this need not be the
case. In the proof of the Chevalley criterion, if ζ is an identity then so is η. So, for a normal 0-fibration,
p~ : ΦE -> p/B has a left adjoint with unit an identity."

Notes:
1. Gray [2] = "Fibred and cofibred categories", La Jolla.
2. I don't know which paper Street means by "the next paper".
3. In a pseudo L-algebra E, with structure morphism c: LE -> E, ζ denotes the isomorphism from Id_E to unit
composed with c. E is normalized if ζ is equality.
4. η is the unit of the adjunction.

But that seems to claim that in Cat it doesn't matter whether you use iso or  equality in the Chevalley
condition. Does that accord with your understanding?

(Continue reading)

Jean Bénabou | 22 Jul 06:24 2014
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Re: Composition of Fibrations

Dear Steve,

At least one ambiguity is solved. Chevalley gave as criterium (for opfibrations) that the arrow 
p~ : ΦE -> p/B  in your mail has a left adjoint with unit the identity.
When the 2-category is Cat this condition is satisfied iff  p is an opfibration which has an opcleavage. The
choice of the adjoint defines the opcleavage.

Let us for the sake of precision call  Street criterium the existence of a left adjoint with unit an iso, and
Street opfibrations (in Cat) the functors which satisfy this condition. 
They need not be opfibrations in the sense of Grothendieck which is almost unanimously adopted. It is
unfortunate to have given them the name of (op)fibrations, not only because of the ambiguity as we have
seen, but because the fibers are meaningless, in particular the fibers over two isomorphic objects of the
base B need not be isomorphic. 

I'm almost sure that Neil Ghani, Richard Garner, Claudio Hermida and Thomas Streicher meant Grothendieck
fibrations, and the genuine Chevalley condition in your answer, as I did.

Regards,

Jean

> Dear Jean,
> 
> Street's result is as follows. The arrow p: E -> B is a 0-fibration over B if and only if the arrow
>  p~ : ΦE -> p/B
> corresponding to the 2-cell
> 
> ΦE  --pd1--> B
> |                      ||
> d0                   ||
(Continue reading)

Mike Stay | 21 Jul 23:41 2014
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Double category question

Consider this setup:

*→*→*→*
↓⇙↓⇙↓⇙↓
*→*→*→*
↓⇙↓⇙↓⇙↓
*→*→*→*
↓⇙↓⇙↓⇙↓
*→*→*→*

What kind of higher category models the case where never have a path
that goes right twice in a row?  There are four paths from the upper
left to the lower right satisfying that condition:
→↓→↓→↓
→↓→↓↓→
→↓↓→↓→
↓→↓→↓→
We can almost do this with a double category:  we take the product of
the points above with {0,1} and then say for horizontal neighboring
points x, x' we have a single morphism
    (x, 0) -R-≥ (x', 1)
and for vertical neighboring points y, y' we have two morphisms
    (y, 0) -D1-≥ (y', 0)
    (y, 1) -D2-≥ (y', 0).
This way it's impossible to form the composition of two arrows going right.

The squares would need to be of the form R;D2 => D1;R, but the types
don't match:  R;D2 goes from (s,0) to (t,0) while D1;R goes from (s,0)
to (t,1).

(Continue reading)

Steve Vickers | 21 Jul 22:06 2014
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Re: Composition of Fibrations

Dear Jean,

Street's result is as follows. The arrow p: E -> B is a 0-fibration over B if and only if the arrow
   p~ : ΦE -> p/B
corresponding to the 2-cell

ΦE  --pd1--> B
|                      ||
d0                   ||
|         pλ  =>   ||
v                     ||
E   --p------> B

has a left adjoint with unit an isomorphism.

Here ΦE = E/E and p/B are comma objects, d0 and d1 are projections, and λ is the canonical 2-cell in a comma
square (in this case for ΦE). 0-fibration is opfibration.

Regards,

Steve.

> On 21 Jul 2014, at 19:02, Jean Bénabou <jean.benabou <at> wanadoo.fr> wrote:
> 
> Dear Steve,
> 
> Thank you for your prompt  answer. 
> 
> Let me first clarify a possible ambiguity. 
> The Street fibrations I was referring to are defined in his paper: 
(Continue reading)

Jean Bénabou | 21 Jul 20:02 2014
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Re: Composition of Fibrations

Dear Steve,

Thank you for your prompt  answer. 

Let me first clarify a possible ambiguity. 
The Street fibrations I was referring to are defined in his paper: 
Fibrations in bicategories.  Cahiers Top. Geom. Diff. 21 (1980)
When the bicategory is Cat, they do not coincide with the usual fibrations. In particular every
equivalence is a Street fibration.

There might be another ambiguity about what you call the Chevalley criterium. Could you please tell me with
precision what it is (I assume p: B -> A is a map in a 2-category C with comma objects and 2-pullbacks)

I shall come back to mathematical questions as soon as these two ambiguities are solved.

Best wishes,

Jean

Le 21 .. 2014 à 14:30, Steve Vickers a écrit :

> Dear Jean,
> 
> Thank you for your detailed comments.
> 
> Something I should say straight away is that the duality argument I had in mind, dualizing 2-cells, might
be OK to deal with left adjoints to reindexing but was completely wrong for right adjoints. Already,
Richard Garner and Claudio Ermida (thanks to both of them) have shown me that it doesn't do the job.
> 
> I also want to stress that at no point did I intend to set up my own definition of fibration. I was following
(Continue reading)


Gmane