Gwen.Salaun | 18 Aug 17:40 2014

Postdoc Position on Verification of Asynchronously Communicating Systems, Inria/LIG, Grenoble, France

The Convecs team (Inria Grenoble Rh?ne-Alpes research center / LIG
laboratory) recruits a postdoc. More information at:

Applications should be addressed directly to Gwen Sala?n, preferably
by e-mail. Applications received after October 10th, 2014 might not be
considered if a candidate has been selected already.

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Maria Manuel Clementino | 18 Aug 17:15 2014

postdoc grants in Portugal

Our national Foundation for Sciences and Technology
accepts applications for postdoctoral grants until September 30, 2014:

The Regulations can be found at  [in Portuguese].

In order to apply the candidate needs the support of a supervisor,
working in a Portuguese university.

If you're interested, please send me an email.

Maria Manuel Clementino
<mmc <at>>

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tholen | 11 Aug 09:01 2014

Book available

Dear ?categories? readers,

This is to draw your attention to the recently published book

?MONOIDAL TOPOLOGY --  A Categorical Approach to Order, Metric , and

edited by Dirk Hofmann, Gavin J. Seal and Walter Tholen. It appeared in
the ?Encyclopedia of Mathematics and Its Applications? series of
Cambridge University Press, as vol. 153; see

This 500-page book gives a rather self-contained introduction to the
subjects mentioned in its title, including category theory itself. The
list of contents below (including chapter authors and lengths of
chapters) may describe best to you what to expect. Largely absent is a
treatment of (Cauchy-Lawvere-type) completeness which, together with
other more advanced themes, is to be treated in a follow-up book.


I      Introduction (Robert Lowen, Walter Tholen; 14pp)
    1   The ubiquity of monoids and their actions
    2   Spaces as categories, and categories of spaces
    3   Chapter highlights and dependencies

II     Monoidal structures (Gavin J. Seal, Walter Tholen; 127pp)
1	Ordered sets
(Continue reading)

Fred E.J. Linton | 6 Aug 08:52 2014

YT at Seoul ICM 2014

It surprises me as much as anyone, but the price of eschewing LaTeX
seems to be to wind up learning a smattering of HTML ... and of PS (!).

Or so I must infer from the nature of the talk I'm giving either to
geometers or to mathematical pedagogues at the upcoming ICM -- entitled
"A piecewise cubic PostScript trefoil", it offers a piecewise polynomial 
parametrization of the trefoil knot -- a continuously differentiable 
stitching together of six rotated and reflected copies of one basic 
pattern-curve arising as part of the graph of a well-chosen cubic 
polynomial that any basic linear algebra student can come up with.

The cute trick is that you don't need any fluency in linear algebra
technique at all -- PostScript will do it all for you -- solve your
problem, plot your cubic curve, and rotate and reflect it, as much as
is required.

For a five-sheet comix-spread PDF (caution: 2+ MB) of the talk slides, grab .

And enjoy, with a nice, cool, mint julep :-) .

-- Fred

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Adam Gal | 5 Aug 17:30 2014

Re: Final objects in 2-categories

Dear Jonathan and David,

Thanks for your answers  and the references. This notion is what
appeared naturally in our setting so it's nice to know that people
have worked on it. We would very much like to hear more about what you
(Jonathan) are working on in this respect.

Sorry for the late letter but I had very bad internet access in the last week.

On Sat, Jul 26, 2014 at 3:40 PM, Jonathan CHICHE 齊正航
<chichejonathan <at>> wrote:
> Dear Adam,
> I don't have much time right now and have not read your paper carefully. However, to elaborate on David
Roberts's answer, in my work about the homotopy theory of 2-categories, the property which I have found
the most useful  (and which shows up in many natural circonstances) is the following. Given  a 2-category A,
let us say that an object z of A has a terminal object if Hom(a,z) has a terminal object for every object a of A.
This terminology was suggested to me by Jean Bénabou. It is of course compatible with the  usual
definition if the 2-category happens to be Cat. It can be shown that, if a small 2-category A admits such an
object, then the map from A to the  point is a weak equivalence, i.e. its nerve is a simplicial weak
equivalence. I have some papers around related stuff, which I could communicate when  they are in their
final version.

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(Continue reading)

David Roberts | 5 Aug 07:44 2014

Strictification avoids Choice?

Dear all,

reading Mac Lane and Pare's JPAA paper on coherence for bicategories,
it seems to me that both the strictification st(B) of a bicategory B
and the (weak) 2-functors st(B) --> B and B --> st(B) don't require
the axiom of choice (my copy of CWM is on loan, else I would check
more of the details -- for monoidal categories -- given there).

Is this true? Or rather, can we prove B <--> st(B) is an equivalence
of bicategories in the sense of having an adjoint biequivalence of
bicategories in the absence of choice? Is there a reference I can
point to, not of this statement, but of a result that justifies this?
Mac Lane and Pare do not seem to go far enough for me to reasonably
conclude this stronger statement.

Best regards,


PS the paper mentions that the coherence theorem for bicategories is a
special case of a result from Bénabou's thesis. I had a skim, but it
didn't leap out at me, and so apologies if everything I need is in
there and I couldn't see it.

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Zhen Lin Low | 4 Aug 20:24 2014

A condition for functors to reflect orthogonality

Dear categorists,

I am wondering if the following property of a functor U : C -> D has a name
in the literature:

* For every lifting problem in C and any solution in D to the image under
U, there is a unique solution in C whose image under U is that solution.

More precisely:

* For any morphisms X -> Y and Z -> W in C, the induced commutative diagram

      C(W, X) ------> C(Z, X) \times_{C(Z, Y)} C(W, Y)
         |                            |
         |                            |
         v                            v
     D(UW, UX) --> D(UZ, UX) \times_{D(UZ, UY)} D(UW, UY)

   is a pullback square.

Of course, any fully faithful functor has the property in question; a less
trivial example is the projection from a (co)slice category to its base.
Every functor between groupoids has this property, so they need not be
faithful. One also notes that the class of functors with this property is
closed under composition.

It is not hard to see that if a functor has the above property, then it
reflects both orthogonality and weak orthogonality in the naive sense. The
converse is false. Nonetheless, my inclination is to call these functors
(Continue reading)

George Janelidze | 4 Aug 16:52 2014

Re: Present and future

Dear Jean,

>> All I was trying to say (more than once) is that all of them, including
>> fibrations, are very important.
> you don't have to convince me of that, except for indexed categories which
> I consider as a VERY BAD approach to fibered ones. I have for years said
> so, WITH MATHEMATICAL ARGUMENTS, which have not convinced you, but seem to
> convince more and more people.

You could not convince me because I agree, and, moreover, I knew that even
before we first met (in Predela, Bulgaria). More precisely, I know, from
your remarks, but also independently, many mathematical examples where using
fibrations is infinitely better than using indexed categories. I only insist
on replacing "always very bad" with "sometimes very bad". To explain why I
say "before we first met", let me mention 'my' example: extending
Inassaridze's work on generalized satellites in early 70s, my first step was
exactly to replace indexed categories with fibrations! By the way, talking
about fibrations, why do we never mention Yoneda's regular spans, as defined

[N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Tokyo 18, 1960,

> Sorry, I shall seem to you very dumb but I don't see much relation between
> left-exact reflections and fibered categories. But you can easily convince
> me if you give many MATHEMATICAL arguments showing the two notions are
> DEEPLY related.

Forgive me, I said "semi-left-exact" (in the sense of
Cassidy--Hebert--Kelly, or, equivalently, one of versions of "admissible" in
(Continue reading)

Vasili I. Galchin | 4 Aug 09:03 2014

Functional programming and William Lawvere's notion of "variable sets"(vs topos of "static sets") in a Topos ...

Dear Cat People ...

      I have been re-reading William Lawvere's description of "variable
sets" (a functor category) in "a elementary topoi" on a discrete
poset/preset vs a "abstract sets" in the topos of abstract sets . In
Haskell( and other FPLs .. ) aren't "pure" computations just over the
topos of abstract sets and "mutable" /"time-varying/stateful"
computations aren't just "variable sets" (i.e. a functor over a
discrete preset-poset)?? Please forgive my germ of an idea

Kind regards,


P.S. Yes I am familiar with Eugenio Moggi's papers on using monads for
stateful computations ..

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Jean Bénabou | 4 Aug 06:33 2014

Re: Present and future

Dear George,

When you say
> I don't want us to live on different planets - so, I am making one more attempt:
I agree with you and could repeat word for word this sentence.

But I disagree totally with you when you say:
> My feeling is that you interpret everything I say as "some kinds of mathematical objects are better than
fibrations" ("some kinds" could be indexed categories, or pseudo-fibrations, or, say, semi-left exact
reflections).And then you give convincing examples where the language of fibrations works better,
I never said, or even hinted, that fibered categories are better than pseudo fibrations or semi-left exact
reflections, but only that they are different and, in particular for semi-left exact reflections that
the analogy was totally superficial.
And I gave many many mathematical arguments to show how radically DIFFERENT they were.

> and then you say that you could not convince me.
These arguments didn't convince you,and I just stated that fact.
> I NEVER said that any of those concepts is better!
I never reproached you that!

> All I was trying to say (more than once) is that all of them, including fibrations, are very important.
you don't have to convince me of that, except for indexed categories which I consider as a VERY BAD approach
to fibered ones. I have for years said so, WITH MATHEMATICAL ARGUMENTS, which have not convinced you, but
seem to convince more and more people.

> Moreover, the relationship between them - which is not exactly an equivalence - is a very serious
mathematical result/discovery/idea,
Sorry, I shall seem to you very dumb but I don't see much relation between left-exact reflections and
fibered categories. But you can easily convince me if you give many MATHEMATICAL arguments showing the
(Continue reading)

Jean Bénabou | 2 Aug 18:00 2014

Present and future

Dear Thomas,

As I told you in my previous private mail you are entitled to have your own view, and to make it public. You
don't have to submit me anything. I shall of course respect your opinion, even if I disagree with it. (by the
way I told exactly the same thing to George Janelidze but, not only I could not convince him, but I had the
impression we were living on different planets!). 
Of course, if I do disagree, I shall tell you why I do, and try to convince you by purely mathematical
arguments, not by the fact that I consider myself as some kind of owner of fibered categories, in spite of
the important developments of this theory which I introduced.
And I promise to study carefully your own arguments,c and to change my views about some questions if you
convince me, mathematically.

This is by no means an an answer to your mail. I am preparing a more ambitious mail, where I shall expose my
views, not only about fibrations but on other important issues, some of which have not, or very little,
been touched by the numerous mails about fibrations exchanged during the last weeks. 
Because of the comprehensive scope of this future mail, I beg you to be patient, i shall need some time.
This future mail shall, in a sense, be addressed to me. I'm getting old, and I need to think a little about what
I have done, and what I should have done. (Not only in mathematics of course, but the other domains are
between me and me).

Best to all,

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