<?xml version="1.0" encoding="UTF-8"?>
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns="http://purl.org/rss/1.0/" xmlns:taxo="http://purl.org/rss/1.0/modules/taxonomy/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:syn="http://purl.org/rss/1.0/modules/syndication/" xmlns:admin="http://webns.net/mvcb/">
  <channel rdf:about="http://blog.gmane.org/gmane.science.mathematics.categories">
    <title>gmane.science.mathematics.categories</title>
    <link>http://blog.gmane.org/gmane.science.mathematics.categories</link>
    <description/>
    <syn:updatePeriod>hourly</syn:updatePeriod>
    <syn:updateFrequency>1</syn:updateFrequency>
    <syn:updateBase>1901-01-01T00:00+00:00</syn:updateBase>
    <items>
      <rdf:Seq>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7733"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7731"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7730"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7729"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7728"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7725"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7723"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7721"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7718"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7717"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7716"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7715"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7713"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7709"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7704"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7703"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7702"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7698"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7696"/>
        <rdf:li rdf:resource="http://comments.gmane.org/gmane.science.mathematics.categories/7689"/>
      </rdf:Seq>
    </items>
    <image rdf:resource="http://gmane.org/img/gmane-25t.png"/>
    <textinput rdf:resource=""/>
  </channel>
  <image rdf:about="http://gmane.org/img/gmane-25t.png">
    <title>Gmane</title>
    <url>http://gmane.org/img/gmane-25t.png</url>
    <link>http://gmane.org</link>
  </image>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7733">
    <title>Double groupoids and crossed modules</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7733</link>
    <description>&lt;pre&gt;Hi,

In clearing out my office at Bangor University (which for two years or
more I have hardly used) I came across the  first version of my work
with Chris Spencer;   this  was rejected by Saunders for JPAA, after two
negative reports (one described it as "good file drawer stuff",  and the
other remarked that part was known to Verdier and not published. So it
was eventually revised into 2 papers, and published elsewhere, in 1976.
I decided to scan it to have it on my files, and in case anyone else was
interested. It contains some accounts of homotopies and  a Whitehead
theorem, not published elsewhere, except as part of work in the mid
1980s with Philip Higgins. So this is now available,  as they say "as
is", on my preprint page,

http://pages.bangor.ac.uk/~mas010/brownpr.html
&amp;lt;http://pages.bangor.ac.uk/%7Emas010/brownpr.html&amp;gt;

which also has a recent  presentation on John Robinson, sculptor.

Ronnie

&lt;/pre&gt;</description>
    <dc:creator>Ronnie Brown</dc:creator>
    <dc:date>2013-05-24T15:18:08</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7731">
    <title>PhD positions in bialgebraic semantics in Warsaw</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7731</link>
    <description>&lt;pre&gt;Two PhD student positions will soon be available to work under the direction
of Bartek Klin in the Institute of Informatics, University of Warsaw.

The positions are in the field of semantics of programming
languages and process algebras, in connection to the project
"Modular operational semantics: a bialgebraic approach", funded
by the Polish National Science Center. The applicants should have
solid background in Mathematics and Computer Science, and
be interested in topics such as semantics of programming languages,
category theory, process algebra, formal methods.

The positions will be available from October 2013. Interested
candidates should apply for a PhD fellowship at the Warsaw Center
of Mathematics and Computer Science (http://www.wcmcs.edu.pl/node/38),
with the deadline of

*** June 7th, 2013 ***.

A successful candidate will earn 5000PLN(~1200EUR)/month
(3500PLN tax-free from the WCMCS fellowship + 1500PLN pre-tax funded
by the research project), subject to yearly evaluation of progress.
The positions are associated with teaching duties of up to 60 hours/year.

Any queries about the positions should be addressed to Bartek Klin,
klin&amp;lt; at &amp;gt;mimuw.edu.pl .


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Bartek Klin</dc:creator>
    <dc:date>2013-05-22T22:01:25</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7730">
    <title>Post-doctoral Position at the University of Cambridge</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7730</link>
    <description>&lt;pre&gt;A post-doctoral position is available at the University of Cambridge
Computer Laboratory on the European Research Council Advanced Grant
ECSYM (Events, Causality and Symmetry---the next generation semantics).
The position is initially for one year, starting after 1 July 2013, with
the possibility of renewal after that period. The position is for a
talented researcher in theoretical computer science or mathematics, with
expertise in several of the areas of games and logic, concurrency,
category theory, type theory and semantics.

Further details on the ECSYM project can be found at
http://www.cl.cam.ac.uk/~cdt25/ecsym/

Details on how to apply can be found on
http://www.jobs.cam.ac.uk/job/-28982/

Application deadline: June 16, 2013


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Pierre Clairambault</dc:creator>
    <dc:date>2013-05-22T10:45:51</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7729">
    <title>samson&lt; at &gt;60, 28-30 May 2013 Full Program</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7729</link>
    <description>&lt;pre&gt;A Conference in Honour of Samson Abramsky on the event of his 60th Birthday

28-30 May 2013

Oxford Department of Computer Science, Lecture Theatre B 

Organisers: Bob Coecke, Luke Ong, Prakash Panangaden
Local organisers: Destiny Chen, Aleks Kissinger  

Please contact Destiny Chen &amp;lt;destiny.chen&amp;lt; at &amp;gt;cs.ox.ac.uk&amp;gt; for all questions.

www: http://www.cs.ox.ac.uk/sa60/index.html

PROGRAM:

Tuesday 28th May »

08.30-09.15 Registration and Payment
09.30-10.00 Introduction/welcome by Head of Department
10.00-10.30 Luke Ong
10.30-11.00 Radha Jagadeesan
             Title:  Linearizability, Revisited.
 
11.00-11.30 BREAK
 
11.30-12.00 Nikos Tzevelokos &amp;amp; Andrzej Murawski
             Title: Towards Nominal Abramsky
12.00-12.30 Paul-Andre Mellies
             Title: Dialogue categories and Frobenius
 
12.30-14.00 LUNCH
 
14.00-14.30 Glynn Winskel
             Title: Quantum event structures and strategies''
14.30-15.00 Marcelo Fiore
             Title: The Algebra of DAGs
 
15.00-15.30 BREAK
 
15.30-16.00 Dusko Pavlovic
             Title: Concepts extended in time
16.00-16.30 Noson Yanofsky
             Title: Kolmogorov Complexity of Categories

16.30-16.45 BREAK
 
16.45-17.15 Marina Lenisa
             Title: Unfixing the Fixpoint: the theories of the lambdaY-calculus
17.15-17.45 Martin Hyland

Wednesday 29th May »

09.30-10.00 Dan Ghica
             Title: Diagrammatic Reasoning for Delay-Insensitive Asynchronous Circuits
10.00-10.30 Ross Duncan
             Title: Tracing a path from game semantics to post-quantum protocols
10.30-11.00 Aleks Kissinger
 
11.00-11.30 BREAK
 
11.30-12.00 Jamie Vicary
             Title: Stitching the Logical Quilt
12.00-12.30 Bob Coecke
             Title: The logic of quantum mechanics - take II

12.30-14.00 LUNCH
 
14.00-14.30 Philip Scott
             Title: A categorical model of higher-order quantum computation
14.30-15.00 Chris Heunen
             Title: On the functor l^2
 
15.00-15.30 BREAK
 
15.30-16.00 Simon Gay
             Title: Deadlock-Free Processes and Non-Zero Vectors
16.00-16.30 Mingsheng Ying
             Title: Floyd-Hoare logic for quantum programs
 
16.30-16.45 BREAK
 
16.45-17.15 Ray Lal
             Title: A sheaf-theoretic approach to the contextuality of cluster states
17.15-17.45 Shane Mansfield
             Title: On the reality of observable properties
 
19.30 Dinner at Wolfson

Thursday 30th May »

09.30-10.00 Achim Jung
             Title: Continuous domain theory in logical form
10.00-10.30 Mike Mislove
             Title: Searching for a Random Variables
10.30-11.00 Prakash Panangaden
             Title: Which is the fairest of them all?  The expressive power of indeterminate primitives.
 
11.00-11.30 BREAK
 
11.30-12.00 Peter Hines
             Title: Coherence in the untyped setting
12.00-12.30 Gordon Plotkin
 
12.30-14.00 LUNCH
 
14.00-14.30 Chris Hankin &amp;amp; Malacaria
14.30-15.00 Jamie Gabbay
15.00-15.30 Jim Laird
             Title: computational interpretations of linear logic
 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Bob Coecke</dc:creator>
    <dc:date>2013-05-22T11:18:28</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7728">
    <title>Call for Papers- special issue APAL</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7728</link>
    <description>&lt;pre&gt;

-------------------------------------------------------------------------
Call for Papers: Fourth Workshop on Formal Topology (4WFTop)
-------------------------------------------------------------------------
Special Issue of Annals of Pure and Applied Logic
-------------------------------------------------------------------------

The Fourth Workshop on Formal Topology was held in Ljubljana in June 2012:

http://4wft.fmf.uni-lj.si/

The proceedings of this workshop will be published as a special issue of
the Annals of Pure and Applied Logic, with the following guest editors:

Thierry Coquand, Maria Emilia Maietti, Giovanni Sambin, Peter Schuster.

These proceedings are open for high-level research papers on topics from
or closely related to formal topology, that is, constructive and/or
point-free topology including its applications and its foundations.

-------------------------------------------------------------------------
Submissions by email to: 4WFTop.apal&amp;lt; at &amp;gt;math.unipd.it
-------------------------------------------------------------------------
Please let us know if you plan to submit a paper as soon as possible
Deadline for submissions: Thursday, 31 October 2013
-------------------------------------------------------------------------





[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Maria Emilia Maietti</dc:creator>
    <dc:date>2013-05-18T08:35:18</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7725">
    <title>CT2013 - early bird registration</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7725</link>
    <description>&lt;pre&gt;Dear all,

This is a reminder that early bird registration for Category Theory 2013
closes TOMORROW, Friday 17th May. After that, the cost of registration goes
up by 10% across the board.

Any registrations submitted by fax, post or other means at a time
verifiably before the 18th in your local timezone will be deemed eligible
for the early bird rates; any submitted thereafter will not.

The registration page may be found at:

http://web.science.mq.edu.au/groups/coact/seminar/ct2013/registration.html

If you believe you have submitted a registration form already, but have
received no confirmation of its receipt, please email me to let me know.

Richard Garner (for the CT2013 organising committee)

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Richard Garner</dc:creator>
    <dc:date>2013-05-16T09:46:08</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7723">
    <title>GlynnFest Workshop, May 31st and June 1st, Cambridge University Computer Laboratory</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7723</link>
    <description>&lt;pre&gt;We are happy to announce a workshop to honour Glynn Winskel on the occasion  of his 60th birthday.

The workshop will take place at Cambridge University Computer Laboratory on  May 31st and June 1st.

The speakers will be:

- Samson Abramsky, University of Oxford
- Henrik R. Andersen, Configit
- Steve Brookes, Carnegie-Mellon University
- Pierre-Louis Curien, University of Paris 7
- Olivier Danvy, University of Aarhus
- Marcelo Fiore, University of Cambridge
- Thomas T. Hildebrandt, IT University of Copenhagen
- Martin Hyland, University of Cambridge
- Kim G. Larsen, University of Aalborg
- Ugo Montanari, University of Pisa
- Mogens Nielsen, University of Aarhus
- Prakash Panangaden, McGill University
- Andy Pitts, University of Cambridge
- Gordon Plotkin, University of Edinburgh
- Vladimiro Sassone, University of Southampton

Participation to the workshop is open, but attendees are kindly requested to register in advance.
More details about the workshop venue and program and about the registration procedure can be found at the following link
http://www.itu.dk/research/models/wiki/index.php/GlynnFestWorkshop

Best wishes on behalf of the organizing committee,
Thomas Hildebrandt


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>hilde&lt; at &gt;itu.dk</dc:creator>
    <dc:date>2013-05-14T09:27:38</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7721">
    <title>Zig Zags?</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7721</link>
    <description>&lt;pre&gt;Dear Toby,

Thank you for your explanations. I really admire your elegant idea to use zig zags instead of 2-pullbacks. 
But I am an old fashioned mathematician and I like precise definitions with which I can (try to) prove precise results.
You suggest to use zig zags. Could you please tell me what are the maps in the zig zags: anafunctors? functors? (in both cases what properties do you assume about them?), equivalences of categories? (in that case in what sense?)
How do you compose your zig zags? You say "directly", do you mean by mere concatenation?
Obviously there would be a huge amount of such zig zags, thus you would probably want to work up to some identification. Could you please tell me, with precision, when two such zig zags between two categories A and B should be identified?
In the case of spans, using 2-pullbacks, what are the maps in your spans, when should two such spans between A and B be identified?
You say, I quote you:

Indeed, so one must also define natural isomorphism of equivalences
If you have any difficulty, the answer is in Makkai's anafunctor paper

Thank you for the tactfulness with which you point out my alleged ignorance, but I happen to know what anafunctors are. I have even given a very simple definition of them. It is still in the nLab, under my name (I have checked that recently)
But due to my limited imagination, I don't know how to use anafunctors here. Could you please tell me how they can be used to decide when two spans or two zig zags are equivalent.
Finally, I cannot help thinking about the poor people who are not familiar with anafunctors. Are they doomed forever to never understand what it means for two categories to be equivalent?
I look forward to your answers.

Very respectfully yours,
Jean 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>majordomo&lt; at &gt;mlist.mta.ca</dc:creator>
    <dc:date>2013-05-13T12:50:57</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7718">
    <title>on a subcategory of algebras for a monad</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7718</link>
    <description>&lt;pre&gt;Hi,

I received the following question from a grad student that I was unable to
answer, but maybe you can (shared with permission). The subcategory Comp_M
he introduces below can equally be defined to be the inverter of the
counit of the monadic adjunction. But I don't see how this universal
property helps understand limits in the subcategory. We suspect a left
adjoint to the inclusion is unlikely.

Can you help? Or have you seen something like this before?

Best,
Emily

***
??
Hi folks,
??
I'm interested in closure properties of a particular subcategory of the
category of algebras of a monad. To be more precise, let C be a locally
presentable category and M be a monad on C. The category of algebras Alg_M
has all limits, and they are computed in C. Denote by Comp_M the full
subcategory of Alg_M of "M-complete objects" (does anyone have a better
name?), with objects those X in C such that the unit X -&amp;gt; MX is an
isomorphism, viewed in the natural way as M-algebras (using the inverse MX
-&amp;gt; X).
??
My question: Is Comp_M closed under (actually: sequential) limits, computed
as limits in Alg_M?
??
For some examples that come to mind immediately, the answer is clearly yes,
because Comp_M is either trivial (e.g., if M is the free monoid monad on
Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting example
is Bousfield-Kan R-completion, for which I don't know the answer.
??
In fact, I'm interested in left exact monads; in this case, the idempotent
approximation is given by the equalizer of the two natural maps M -&amp;gt; M^2,
but I'm not sure if this is relevant. What I'm hoping for is a sufficient
criterion or a good counterexample in the abstract situation.
??
Many thanks!



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Emily Riehl</dc:creator>
    <dc:date>2013-05-10T13:05:44</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7717">
    <title>(In)accessible comonads and (non)Grothendieck toposes</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7717</link>
    <description>&lt;pre&gt;Hi all,

I am just wondering where it was first stated (for both directions) that the category of coalgebras for a comonad on a Grothendieck topos E is again Grothendieck if and only if the underlying endofunctor of E is accessible. 

A modern argument might go as: the topos of coalgebras is Grothendieck if and only if it is locally presentable if and only if the endofunctor is accessible, the original probably just mentioned preservation of filtered colimits.

Many thanks,

David Roberts

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>David Roberts</dc:creator>
    <dc:date>2013-05-09T03:07:21</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7716">
    <title>Reminder: CT2013 abstracts</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7716</link>
    <description>&lt;pre&gt;Dear all,

This is a reminder that if you wish to take advantage of the extended 1
June deadline for submitting abstracts to CT2013, you are requested to
email me **by 10 May** saying that you intend to submit an abstract.
This is to help with planning.

Abstracts received after 10 May will not be accepted unless you have
emailed before then.  I am acknowledging all such emails, so if you have
written to me and not received a reply, there has probably been a
technological problem - in which case, please try again.

Best wishes,
Tom


&lt;/pre&gt;</description>
    <dc:creator>Tom Leinster</dc:creator>
    <dc:date>2013-05-08T14:55:07</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7715">
    <title>Samuel Eilenberg Centenary Conference - Second Announcement</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7715</link>
    <description>&lt;pre&gt;-------- SECOND ANNOUNCEMENT-------------------------------------------------

Dear Colleagues,
below is the Second Announcement of the Samuel Eilenberg Centenary Conference
which will be held in Warsaw, July 22-26, 2013. The organizers cordially
invite you to participate in the event commemorating one of the founders of
our field.

-------- SCIENTIFIC PROGRAM--------------------------------------------------

Ten plenary speakers confirmed their attendance and some titles and abstracts
of their lectures are already posted at:

http://eilenberg100.ptm.org.pl/programme

All participants are welcome to propose contributed talks by filling an
appropriate entry in the registration form and uploading an abstract. Note
that the deadline for submissions of titles and abstracts of the contributed
talks is May 31, 2013.

-------- REGISTRATION--------------------------------------------------------

The organizers of the conference cordially invite you to register for the
meeting at:

http://eilenberg100.ptm.org.pl/registration

The conference fee is 300 PLN (approx 70 Euro or 100 USD), and it will raise
to 400 PLN on June 1, 2013.

The fee can be paid by credit card on-line or a bank transfer. For details
see: http://eilenberg100.ptm.org.pl/registration

-------- ACCOMMODATION-------------------------------------------------------

There are several accommodation options in a walking distance from the
conference venue. In some hotels the organizers pre-booked rooms for the
conference participants. More information at:
http://eilenberg100.ptm.org.pl/accommodation .

Visit the conference website eilenberg100.ptm.org.pl for updated information.
If you have any questions to the organizers ask them via contact form at the
website.

We are looking forward to seeing you in Warsaw.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Marek Zawadowski</dc:creator>
    <dc:date>2013-05-08T09:34:48</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7713">
    <title>PhD program in Warsaw</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7713</link>
    <description>&lt;pre&gt;Dear Categorists,

Warsaw Center of Mathematics and Computer Science (WCMCS) has opened
registration for PhD programs in mathematics and computer science at the
University of Warsaw and Mathematical Institute of Polish Academy of
Science. A number of PhD scholarships financed by WCMCS will be awarded to
the most promising graduate students. The details of the program are
available at wcmcs.edu.pl/projects. You can apply on-line at
http: //wcmcs.edu.pl/submit-application/admissions-fellowships
The deadline for application is June 7th, 2013.

Best regards,
Marek Zawadowski


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Marek Zawadowski</dc:creator>
    <dc:date>2013-05-07T21:05:22</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7709">
    <title>"Terminolgy" re-visited</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7709</link>
    <description>&lt;pre&gt;Dear all,

I cannot type any form of LaTeX, and do not know the "standard" ways to introduce indices,exponentials and so on, using only the typing which is admitted on this list. Thus I shall use "non standard" notations, very simple, which I shall explain precisely.
If A is a category an object a of A can be identified with a functor "name of a" which I denote by "a": 1 --&amp;gt; A .
If  F: A --&amp;gt; C and G: B --&amp;gt; C are functors I denote by  F/G the comma category they define, and by F//G their
  2-pull-back sometimes called their pseudo pull-back.
I shall call "weak equivalence" a functor F: A --&amp;gt; B  full and faithful and essentially surjective (ff-es) and say that A is weakly equivalent (we) to B if there is such an F. This defines  a preorder relation which I denote by 
W(A,B). It is symmetric iff the Axiom of choice (AC) holds.
A strong equivalence between  A and B is a pair of adjoint functors  F: A --&amp;gt; B  and  F': B --&amp;gt; A  such that the adjunction morphisms are isos. I shall say that  A and B are strongly equivalent if such a pair exists and denote by  SE(A,B) this equivalence relation.
If S is a category I denote by Fib/S the 2-category of fibrations over S
I denote by S° the dual of S . An S indexed category C is a pseudo functor   C: S° --&amp;gt; Cat .  For each map  
f: s --&amp;gt; t  of  S , I denote by f*: C(t) --&amp;gt; C(s) the value of  C on f. If g: t --&amp;gt; u is another map of S, I denote by  c(f,g) the isomorphism f*g* --&amp;gt; (gf)*. I shall assume C normalized i.e. for each object  s of S, id(s)* = id(C(s))
For an indexed functor  F: C --&amp;gt; C'  i use the following notations: If s is an object of S, F(s): C(s) --&amp;gt; C'(s) , 
if  f: s --&amp;gt; t is a map of S  is the iso of functors  F(s)f* --&amp;gt; f*F(t)
(I'm aware of the ambiguity of denoting by the same  f* the "re-indexing functors" defined by C and C' but it is the best I can do with the limited typographic means I use, without having too cumbersome notations) 
If S is has pull-backs, I denote by Cat(S) the category of categories internal to S

 
Answer to Toby Bartels:
(i) You propose to compose the spans by pullbacks. Here is an example of two spans  A &amp;lt;-- X --&amp;gt; B and 
B &amp;lt;-- Y --&amp;gt; C such that the functors  A &amp;lt;--X  and  Y --&amp;gt; C are isos, the functors X --&amp;gt; B and  B &amp;lt;-- Y have unique quasi inverses, and if  Z is the pullback none of the functors  A &amp;lt;-- Z  and  Z --&amp;gt; B is a weak equivalence:
Take A=X=Y=C=1, take for B the coarse category with two objects  a and b  , for X --&amp;gt; B and  B &amp;lt;-- Y  the functors  "a": 1 --&amp;gt; B  and  B &amp;lt;-- 1: "b"  . The pullback  Z is 0 
How stronger a counter example do you need? And using zig-zags will make the situation even worse.
(ii) you define equivalences by spans, not "up to anything". With this definition an equivalence between 1 and 1 is any non empty coarse category. Every non empty set determines up to isomorphism such a category. thus there are at least as many equivalences from 1 to 1 in your sense as there are non empty sets. A bit much don't you think?

Answer to David Roberts. I quote you:
"Also, for the purposes of further discussion, the terminology "evil" has been demoted to a mere footnote at the nLab, as it was probably always meant to be by its coiners, and replaced by the morally neutral and more informative name 'principle of equivalence'. Interestingly, Voevodsky's Univalence Axiom is a way of ensuring the principle is always respected (and without awkward acrobatics)."

I am a very thorough person, so I have looked at the instances of "evil" in nLab and found hundreds of them, not mere footnotes.
I shall concentrate on the article: "Grothendieck fibrations in nLab". It has been revised by Mike Shulman on october 11, 2012, i.e. not very long ago, and it has many interesting features. I shall comment only those of them which have some bearings to the present discussion.
Although it is supposed to deal with Grothendieck's fibrations there is a long paragraph,almost a whole page, with the title: "Non evil version". The very first line is: "There is something evil about the notion of fibration.." and the word "evil" is even underlined.
Another interesting feature is the definition of internal fibration in a strict 2-category K (why "strict"? a 2-category is a 2-category. Probably because the notion of 2-category is unbearably evil).
If you use this definition for K = Cat, it is easy to see that such internal fibrations are Grothendieck fibrations equipped with a cleavage. The situation is even worse if K = Cat(S) 
Last but not least, I recommend reading the deep discussion between Shridar Ramesh and Mike Shulman about "indexing" the fibration  Z --&amp;gt; Z/2Z. (but hurry, because after this mail there might very well be a quick revision of the article).
I happen to like very much Grothendieck's fibrations, this would probably qualify me as an evil person, let me say that i'm proud of this kind of evilness.  

Preliminary answer to Marta Bunge:
Many thanks for your mail. Before I give a complete answer, I have some questions:
(1) - What do you mean precisely by a full, faithful essentially surjective indexed functor. Does this notion depend on the indexings?(this is why I took the pain to give precise notations for such indexed functors)
(2) - I suppose that if C is internal to S, your [C] is the canonical (split) indexing defined by [C](s) = S(s,C)  thus your answer to (1) will tell me what you mean by [C] and [D] are weakly equivalent.

Before getting your answers, I can already make a few comments about your mail.
(a) Assuming you have a notion of equivalence E(A,B) reflexive transitive and symmetric, your answer to (ii) is a bit surprising. Given any A, suppose someone asks how many B's are equivalent to A? If B and B' are such B's, they are equivalent, hence if you work up to equivalence, the answer is 1. What do stack completions have to do with this answer? 
(b) You say that you could have worked with fibered categories instead of indexed ones, and I believe you.
But why didn't you do it? What is at stake in this this discussion is the equivalence of categories with or without AC. You know that the relation between indexed categories and fibered ones depends essentially on AC.
I have seen very strange things about this relation.
The strangest one can be found in the Elephant: p.4, one can read:
"we  should make the smallest possible demands on the metatheory within which we interpret the theory of categories (and in particular we shall not assume that it satisfies any form of the axiom of choice)"
True to this very strong statement, the author gives the proof, due to Grothendiek, that indexed categories are "the same thing" as fibered categories equipped with a cleavage. And that is O.K. for me.
But then, without any transition or warning he starts saying:
"Let  P: C --&amp;gt; S be a fibration and CC be the associated indexed category .."
It is not an accidental slip. It can be found in Example 1.3.13, Lemma 1.4.1, Lemma 1.4.5, Lemma 1.4.9, Lemma 1.4.10, Lemma 1.4.11, Theorem 1.4.12, etc..
He does not say "an" associated, but " the" associated. This means not only that every fibration has a cleavage (which is equivalent to AC when restricted to small fibrations) but also comes equipped with a cleavage, which is equivalent to huge form of AC for classes. Not bad considering the strong statement I have quoted.
But even assuming this huge AC, "the" associated indexed category is astounding.
It well known, and was remarked by Grothendieck more than 50 years ago, that a group homomorphism 
P: E --&amp;gt; B is a fibration iff it is surjective, and that a cleavage is a section S of (the underlying map of) P.
You won't find this fact in the Elephant. "The" associated indexed category means that such a P comes equipped with a section.
Imagine the reaction of the students if, in order to explain periodic functions via the surjection P: R --&amp;gt; R/Z , you had to assume that a section S of P is already given, and that S has any influence on the notion of periodic function!
I believe that indexed categories have been a disaster for the mathematicians, some of whom very serious, who have adopted them.
Let me give two more arguments.
(a) They forbid the study of functors P: E --&amp;gt; B more general than fibrations, but which keep enough of the features of the fibrations to be able to prove about them many of the results of fibered category theory. The simplest case is pre-fibrations.
(b) The theory of fibrations is a first order theory, therefore they can be internalized, e.g. in a topos (this is much too strong) Can anybody give a definition of internal indexed categories and functors?To

Excuse me for such a long mail, there are many more things to say about equivalences of categories, but they will have to be postponed to another mail.
 














(a) You work with S-indexed or 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Jean Bénabou</dc:creator>
    <dc:date>2013-05-07T08:23:34</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7704">
    <title>Foundations of Health Information Engineering and Systems (FHIES 2013) - extended submission deadline</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7704</link>
    <description>&lt;pre&gt;THIRD AND FINAL CALL FOR PAPERS

Third International Symposium on
Foundations of Health Information Engineering and Systems
http://www.cs.ox.ac.uk/FHIES2013/

International Institute for Software Technology
United Nations University, Macau
21st-23rd August, 2013


EXECUTIVE SUMMARY - CHANGES SINCE PREVIOUS CALL

Submission deadline extended by two weeks, to May 20th.
Additional category of extended abstracts (max 2 pages) solicited.
Apologies for duplication.


BACKGROUND

ICT plays an increasingly enabling role in addressing the global challenges of healthcare, in both the developed and the developing world. The  use of software in medical devices has caused growing concerns in relation to safety and efficacy. The increasing adoption of health information systems provides great potential benefits but also poses severe risks, both with respect to security and privacy and in regard to patient safety. Hospital and other information systems raise important issues of workflow support and interoperability. Regulators, manufacturers and clinical users have pointed out the need to research sound and science-based engineering methods that facilitate the development and certification of quality ICT systems in health care. Such methods may draw from or combine techniques from various disciplines, including but not limited to software engineering, electronic engineering, computing science, information science, mathematics, and industrial engineering.


AIMS

The purpose of the symposium series on Foundations of Health Information Engineering and Systems is to promote a nascent research area that aims to develop and apply theories and methods from a variety of disciplines for the purpose of modeling, building and certifying software-intensive ICT systems in healthcare. A particular objective of FHIES is to explicitly include a focus on healthcare ICT applications in the developing world (in addition to systems used in the developed countries), since unique engineering challenges arise in that special setting. Because  humans often play a pivotal role in the process of using such systems,  theories from the human factors engineering community may need to be integrated with methods from the technology-oriented domains in order to  create effective engineering methodologies for socio-technical systems  in the healthcare domain. Previous FHIES symposia were held in 2011, in Mabalingwe, South Africa (with post-conference proceedings in Springer LNCS 7151, and in 2012, in Paris, France (with post-conference proceedings to appear in Springer LNCS).


SCOPE

FHIES seeks contributions from both the solution domain (engineering methods) and the problem domain (healthcare and health informatics). Solution-domain papers should present their methods in the context of a concrete application in healthcare, while problem-domain papers should be devised to educate the methods community about unique challenges and characteristics of the healthcare domain. Submissions should seek to inform and further the development, adaptation, evaluation and adoption of formally based and rigorous engineering methods in health care systems.  Topics of interest include but are not limited to:

* modelling, analysis, simulation and verification in health informatics;
* design and verification techniques for software-based ICT and software-intensive medical devices;
* application and integration of foundational methods from different disciplines in engineering and science to health informatics;
* specific engineering challenges of ICT-based health service delivery in different settings, especially in the developing world.

For a more detailed list of topics, see the symposium website.
 

CATEGORIES

We solicit high quality full submissions in the following categories:

* original research contributions (16 pages max)
* application experience, case studies and software prototypes (16 pages max.)
* surveys, comparisons, and state-of-the-art reports (16 pages max.)
* position papers identifying challenges and milestones of a research project (8 pages max.)

We also invite short submissions for special sessions:

* student papers on work in progress on an MSc or PhD project (4 pages max.)
* tool demonstrations (2 pages max.)
* proposals to organize birds-of-a-feather sessions or panels (2 pages max.)
* extended abstracts (2 pages max.)


SUBMISSIONS

Submissions should be in English, prepared in the LNCS format (http://www.springer.de/comp/lncs/authors.html), and all page limits are measured in this format.  Full submissions (those in the first four categories  above) will be judged on the basis of originality, contribution to the  field, technical and presentation quality, and relevance to the symposium; student papers will be judged on clarity of description and the promise of interesting results; tool demonstrations, BOF proposals, and extended abstracts will be judged on relevance to the symposium. All papers will be peer-reviewed by at least three program committee members.   Papers should be submitted via EasyChair, at

   http://www.easychair.org/conferences/?conf=fhies2013

Submission constitutes a commitment for at least one author to attend the symposium and present the paper, if it is accepted.


PUBLICATION

All accepted submissions will be distributed in a technical report at the Symposium. After the event, postproceedings will be published in Springer LNCS. Authors of all accepted full submissions will be invited to  revise their papers, in order to resolve any larger issues raised during reviewing. Authors of accepted short submissions will be invited to submit full papers for review and LNCS publication too.  In addition, a  special issue of a suitable journal is planned, focusing on the overall objectives of FHIES: this will have an open call for contributions.


IMPORTANT DATES

Submission deadline: extended to May 20th
Notification of acceptance: June 12th
Delivery of preproceedings version: July 17th
Symposium: August 21st-23rd
Submission for postproceedings review: October 4th
Notification of acceptance: October 11th
Camera ready version: October 18th
Publication of proceedings: December 23rd


ORGANIZERS

General chairs:

* Zhiming Liu, United Nations University, MO
* Jens Weber, University of Victoria, CA

Programme chairs:

* Jeremy Gibbons, University of Oxford, UK
* Wendy MacCaull, St. Francis Xavier University, CA

Programme committee:

* Ime Asangansi, University of Oslo, NO
* Tom Broens, Mobihealth, NL
* Lori Clarke, University of Massachusetts, US
* David Clifton, University of Oxford, UK
* Gerry Douglas, University of Pittsburgh, US
* Johannes Faber, IIST, United Nations University, MO
* Jozef Hooman, Embedded Systems Institute and Radboud University Nijmegen, NL
* Michaela Huhn, Technische Universität Clausthal, DE
* Shinsako Kiyomoto, KDDI R&amp;amp;D Laboratories Inc, JP
* Craig Kuziemsky, University of Ottawa, CA
* Yngve Lamo, Bergen University College, NO
* Insup Lee, University of Pennsylvania, US
* Orlando Loques, Instituto de Computação, Universidade Federal Fluminense, BR
* Gilbert Maiga, Makerere University, UG
* Dominique Mery, Université de Lorraine, LORIA, FR
* Deshendran Moodley, University of KwaZulu-Natal, ZA
* Jun Pang, University of Luxembourg, LU
* Manfred Reichert, University of Ulm, DE
* Ita Richardson, Lero, University of Limerick, IE
* David Robertson, University of Edinburgh, UK
* Christopher Seebregts, Jembi Health Systems / Medical Research Council, ZA
* Bo Song, Qingdao University of Science and Technology, CN
* Alan Wassyng, McMaster University, CA

Keynote speakers:

* Joe Cafazzo, Centre for Global eHealth Innovation, CA
* Jane Liu, Academia Sinica, TW
* Bill Thies, Microsoft Research, IN




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Jeremy.Gibbons&lt; at &gt;cs.ox.ac.uk</dc:creator>
    <dc:date>2013-05-03T14:55:13</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7703">
    <title>Professorship in Mathematics with focus on Geometry (Stockholm University)</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7703</link>
    <description>&lt;pre&gt;

This position at our department may be of interest to some readers of this list. 
(For instance the geometry research in Sergei Merkulov's group has a quite 
categorical flavour.)

Professor in Mathematics with focus on Geometry

Deadline for application: August 15, 2013.

The Department of Mathematics at Stockholm University with its excellent research record and long lasting traditions has acquired a very special place in Scandinavian mathematics. Professors and lecturers working at the department possess competence in different areas of mathematics ranging from algebra, geometry and analysis to mathematical logic and combinatorics. Geometry with its central role in mathematics has shaped the department’s profile and unites different research groups. We are looking for a mathematician with outstanding research record working in geometry in a broad sense.


http://www.su.se/english/about/vacancies/lecturers-researchers/professor-in-mathematics-with-focus-on-geometry-1.123282

http://eims.ams.org/jobs#/detail/5254388/1,false

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Erik Palmgren</dc:creator>
    <dc:date>2013-05-03T13:36:21</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7702">
    <title>Internal truth objects</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7702</link>
    <description>&lt;pre&gt;Dear categorists,

A standard method for doing logic in category theory is, as far I as
know, by imposing more and more conditions on the subobject posets
Sub(X), which are equivalence classes of monomorphisms with X as the
codomain.

By adding more and more assumptions of this kind one gets more and more
powerful internal logics. The connection between predicates and
functions does not arise in all these stages as long as one does not
request existence of the subobject classifier (therefore almost
certainly turning the whole thing into a topos).

What I am wondering about is whether an alternative approach have ever
been developed, namely by postulating an internal truth-value object
Omega, by introducing predicates as exponentials Omega^X, and so forth.

I would expect, if such an idea have ever been developed it would yield
one of the many ways to provide foundations to the topos theory. The
fact that is not well-known sort of indicates that it is probably not
such a robust idea, in which case I would be grateful if someone would
explain me why.

The only approach to topos theory I know, which might be relevant here
is the one by allegories. Starting from a category C with a
distinguished object Omega (say, internal Heyting algebra) we can
introduce an allegory having the same object as C and as morphisms A-&amp;gt;B
those of C having form A x B -&amp;gt; Omega. Antiinvolution would then follow
by swapping the arguments, and the intersection can be made deducible
from the axiomatic structure of Omega. The question is, of course, the
modular law, and the conditions under which it would follow. Has anyone
studied that?

I would be very grateful for any hints and/or references on the subject.

Best regards,

&lt;/pre&gt;</description>
    <dc:creator>Sergey Goncharov</dc:creator>
    <dc:date>2013-05-03T15:46:05</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7698">
    <title>LICS 2013 - Call for Participation</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7698</link>
    <description>&lt;pre&gt;***********************************************************************
28th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2013)

June 25-28, 2013
(with pre-conference tutorials on June 24)

New Orleans, USA

Call for Participation
http://lii.rwth-aachen.de/lics/lics13/
***********************************************************************

The twenty-eighth ACM/IEEE Symposium on Logic In Computer Science (LICS 2013)
will be held in New Orleans in colocation with MFPS (Mathematical Foundations
of Programming Semantics) and CSF (IEEE Computer Security Foundations).

* DATES: MFPS (June 23-25), LICS (June 25-28), CSF (June 26-28).

* REGISTRATION is now open for all three conferences.

   Please visit http://lii.rwth-aachen.de/lics/lics13/ and follow the
link to Registration.
   The early registration deadline is May 22, 2013.

* LICS SCHEDULE

   - TUTORIALS
     LICS'13 will kick off with tutorials by Hubert Comon
     and Jan Rutten (with MFPS) on Monday 24 June.

   - TECHNICAL PROGRAMME
     The technical programme, from Tuesday 25 to Friday 28 June,
     will open with a special joint session with MFPS XXIX to mark
     the 80th birthday of Dana Scott, with invited speakers:
     Andrew Pitts, Steve Awodey, Andrej Bauer, Robert Harper, and
     Dana Scott. In addition, LICS'13 will feature invited lectures
     by Rajeev Alur, Joseph Halpern (with CSF), Nancy Lynch and
     Prakash Panangaden.

* AFFILIATED WORKSHOPS (June 28-29)

   Foundations of Computer Security (FCS)
      http://prosecco.gforge.inria.fr/personal/bblanche/fcs13/
   Higher-Order Program Analysis (HOPA)
      http://hopa.cs.rhul.ac.uk
   Syntax and Semantics of Low-Level Languages (LOLA)
      http://research.microsoft.com/en-us/events/lola2013/
   Natural Language and Computer Science (NLCS)
      http://www.indiana.edu/~iulg/nlcs.html

* ACCEPTED PAPERS
   http://lii.rwth-aachen.de/lics/lics13/accepted.html

* LICS TEST-OF-TIME AWARDS (LICS 93)
   The Awards Committee consisting of Prakash Panangaden (chair),
   Jean-Pierre Jouannaud, Martin Grohe and Tom Henzinger decided
   to honour the following three outstanding papers from LICS'93
   (held in Montreal, Quebec, Canada):

   - Leo Bachmair, Harald Ganzinger and Uwe Waldmann
     Set constraints are the monadic class,

   - Andre Joyal, Mogens Nielsen and Glynn Winskel
     Bisimulation via open maps,

   - Benjamin C. Pierce and Davide Sangiorgi
     Typing and subtyping for mobile processes.

   The awards will be presented in New Orleans.



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Andrzej Murawski</dc:creator>
    <dc:date>2013-05-02T23:25:20</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7696">
    <title>announcement</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7696</link>
    <description>&lt;pre&gt;Dear Colleagues,

I am pleased to inform you that George Janelidze has been just elected
Academician of the Georgian National Academy of Sciences.

With best wishes

Hvedri Inassaridze



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Hvedri Inassaridze</dc:creator>
    <dc:date>2013-05-03T05:22:39</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7689">
    <title>CT2013 abstracts</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7689</link>
    <description>&lt;pre&gt;Dear all,

The original deadline for submitting abstracts to Category Theory 2013
(Macquarie University, Sydney) was 1 May.

This has now been extended to 1 June.  However, if you have not already
submitted an abstract and intend to do so, please email me by 10 May
declaring your intention to submit.

Abstracts themselves should be submitted via the conference website,

    http://web.science.mq.edu.au/groups/coact/seminar/ct2013/

Best wishes,
Tom Leinster (on behalf of the scientific committee)




&lt;/pre&gt;</description>
    <dc:creator>Tom Leinster</dc:creator>
    <dc:date>2013-05-01T20:29:23</dc:date>
  </item>
  <item rdf:about="http://comments.gmane.org/gmane.science.mathematics.categories/7688">
    <title>Terminology:  Remarks</title>
    <link>http://comments.gmane.org/gmane.science.mathematics.categories/7688</link>
    <description>&lt;pre&gt;Dear all,

Let me first thank the persons who have answered my questions on terminology.
Since the answers were different, it seems that the terminology is not standard and that my questions made sense.

Preliminary remarks. 
By now, everybody understands what kind of categories I was talking about. they are very simple, one might be tempted to say trivial. But in many highly non trivial questions they appear either as "building bricks" of more complex constructions (see e.g. fibrations such that all the fibers are of that kind), or as special cases, unavoidable, of more general situations (e.g. Freyd's notion of "equivalence kernels").
Of course such categories can be "internalized ", say in a topos, (this is much too strong), and it would be nice that the terminology should fit also the internal case, and in particular any reference, explicit or implicit, to AC should be avoided.

1- Discrete versus indiscrete, or coarse, etc. The categories  0 and 1 are both discrete and indiscrete So each name night pose a problem. The terminology "discrete" is by now well established. Thus I think one should avoid "indiscrete" and use "coarse" instead. Thus 0 and 1 are discrete and coarse and that is admissible. In the internal case every sub object of 1 is both discrete and indiscrete. So "subterminal" is a nice name to cover both cases.

2- "Essentially". One has to be careful about this word. It seems to mean "up to equivalence". But that depends on what you call "equivalence of categories". There is a very strong 2-categorical notion, namely a pair of functors  f: A --&amp;gt; B and g: B --&amp;gt; A  with fg and gf isomorphic to identities (with or without the adjunction axioms). But it is useless for our purpose. The one which might serve here is f full and faithful and essentially surjective. But unless we have AC it is not symmetric, even for A and B small. 
Thus with AC we might adopt the suggestions of Thomas and use essentially discrete and essential sub terminator. But do we really need AC or even any notion of equivalence at all?

3- Elementary remarks.
Let S be a category with finite limits. I want to "internalize" the two notions for which I'd like a name, suitable not only when S=Set, but in general.
(i) if X is an internal category I denote by Ob(X) and Map(X) the objects of objects and maps of X and by
d: Map(X) --&amp;gt; Ob(X)xOb(X)  with projections Dom and Codom. When S=Set if f: x --&amp;gt;y is a map of X, df is the "direction" of f.
X is a preordered object iff d is a mono. 
It is easy, using the multiplication of X, to express in terms of finite limits, the fact that X is a groupoid. Hence equivalence relations are definable in any category with finite limits, although composition of arbitrary relations is possible only when S is regular. Moreover if F: S --&amp;gt; S' is a functor which preserves finite limits and   X is an equivalence relation so is F(X)
No essentiality no AC is needed
(ii) the functor  X --&amp;gt; 1 is full and faithful iff the direction map d is an iso. This again is preserved by finite limits preserving functors F.
(iii) If F preserves finite limits and is faithful it reflects equivalence relations and property (ii). In particular the Yoneda embedding preserves and reflects the previous properties. this enables us to work with them as if S=Set. No AC is involved .

For all these reasons I'm not too keen about using "essentially"

I have many more remarks about Fred Linton's and David Robert's postings but this mail is already a bit long. So I shall wait a few days before i send another mail. By that time I hope to have some reactions to the present posting and I shall do my best to answer them.

Best regards, Jean













[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

&lt;/pre&gt;</description>
    <dc:creator>Jean Bénabou</dc:creator>
    <dc:date>2013-05-01T05:17:16</dc:date>
  </item>
  <textinput rdf:about="http://search.gmane.org/?group=$group=gmane.science.mathematics.categories">
    <title>Search Engine</title>
    <description>Search the mailing list at Gmane</description>
    <name>query</name>
    <link>http://search.gmane.org/?group=$group=gmane.science.mathematics.categories</link>
  </textinput>
</rdf:RDF>
