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David Spivak | 1 Feb 01:03
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question about discrete op-fibrations

Hi all,

Here's a quick question perhaps someone here can answer easily.

Let DOF denote the category whose objects are small categories C,D,
etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D.
For a category C, let DOF_{C/} denote the coslice over C.

Question: Does there exist a terminal object in DOF_{C/}?

Thanks!
David

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Mark Weber | 2 Feb 00:29

Re: question about discrete op-fibrations

Dear David

I'll assume by the coslice DOF_{C/} you mean the category whose objects are discrete opfibrations C --> D,
and whose arrows are strictly commuting triangles under C.

In that case the answer to your question is no. When C is empty, DOF_{C/} is  just your category DOF, of small
categories and discrete opfibrations between them, and DOF lacks a terminal object. For suppose that D is
terminal in DOF. Then for any set X, there is a discrete opfibration I(X) --> D, where I(X) is the category
obtained from X by freely adding an initial object. That is, the objects of I(X) are the elements of X
together with one additional object 0, and one has a unique arrow 0 --> x for all x in X.

If F:I(X) --> D is a discrete opfibration, then F(0) is an object of D such that the cardinality of the set of
all arrows with source F(0) is that of X. Thus since D is terminal in DOF, for any set X there is an object x of D
such  that the cardinality of the set of all arrows with source x is that of X. This contradicts the smallness
of D.

With best regards,

Mark Weber

On 01/02/2012, at 11:03 AM, David Spivak <dspivak <at> gmail.com> wrote:

> Hi all,
> 
> Here's a quick question perhaps someone here can answer easily.
> 
> Let DOF denote the category whose objects are small categories C,D,
> etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D.
> For a category C, let DOF_{C/} denote the coslice over C.
>
(Continue reading)

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David Spivak | 2 Feb 06:51
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Re: question about discrete op-fibrations

Hi Mark,

Nice work; thank you for the simple answer and good explanation.

I hope this isn't annoying, but what if I change the problem somewhat
and take DOF(C) to be the full subcategory of Cat_{C/} spanned by the
discrete opfibrations C-->D? Again I want to know whether DOF(C) has a
terminal object. Under this definition, by setting C=empty-category we
get DOF(C)=Cat, which does have a terminal object.

Thanks,
David

On Wed, Feb 1, 2012 at 5:29 PM, Mark Weber
<mark.weber.math <at> googlemail.com> wrote:
> Dear David
>
> I'll assume by the coslice DOF_{C/} you mean the category whose objects are discrete opfibrations C --> D,
and whose arrows are strictly commuting triangles under C.
>
> In that case the answer to your question is no. When C is empty, DOF_{C/}  is just your category DOF, of small
categories and discrete opfibrations between them, and DOF lacks a terminal object. For suppose that D is
terminal in DOF. Then for any set X, there is a discrete opfibration I(X) --> D, where I(X) is the category
obtained from X by freely adding an initial object. That is, the objects of I(X) are the elements of X
together with one additional object 0, and one has a unique arrow 0 --> x for all x in X.
>
> If F:I(X) --> D is a discrete opfibration, then F(0) is an object of D such that the cardinality of the set of
all arrows with source F(0) is that of X. Thus since D is terminal in DOF, for any set X there is an object x of  D
such that the cardinality of the set of all arrows with source x is that  of X. This contradicts the smallness
of D.
(Continue reading)

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Thorsten Palm | 2 Feb 11:22
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Re: question about discrete op-fibrations


David Spivak hat am 31.01.12 geschrieben:

>
> Let DOF denote the category whose objects are small categories C,D,
> etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D.
> For a category C, let DOF_{C/} denote the coslice over C.
>
> Question: Does there exist a terminal object in DOF_{C/}?

Answer: No.

The "op-" is irrelevant; let us look at DF_{C/} instead. There is not
even a weakly terminal object, for the same reason for which there
isn't one in DF itself. (Suppose (T,z) is one. Let A be an arbitrary
small category having a terminal object a. From the object (C+A,incl)
of DF_{C/} we get a discrete fibration f : A-->T. But then for t =
f(a) the slice category T_{/t} is isomorphic to A. There are not
enough t to accommodate all possible A.)

Thorsten Palm

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Thorsten Palm | 3 Feb 15:07
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Re: question about discrete op-fibrations


[ To all, moderator in particular, and with apologies to David:
   Please ignore my previous message. It contains a seriously wrong
   piece of information. ]

Dear David,

This time the answer is "yes" for discrete C, "no" otherwise.

If C is discrete, the functor !_C : C-->1 is a discrete
op-fibration, so that the terminal objet (1,!_C) of Cat_{C/}
belongs to DOF(C).

Now let C contain a non-trivial morphism u : c_0-->c_1 and suppose
that (T,z) is terminal in DOF(C). Construct a category C_{c_1} by
freely adding to C an object d and a morphism v : d-->c_1. Then we
have an inclusion j : C-->C_{c_1}, which is a discrete op-fibration.
We obtain two morphisms (C_{c_1},j)-->(T,z) in DOF(C) by applying z on
the subcategory C and sending v to z(u) and the identity of z(c_1),
respectively. By terminality of (T,z) they have to be the same. But
since z is an op-fibration, z(u) cannot be an identity ---
contradiction.

(I hope Mark hasn't beaten me to it again.)

Thorsten

David Spivak hat am 01.02.12 geschrieben:

>
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ams | 4 Feb 13:52
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CMCS 2012: Call for Participation and Short Contributions

------------------------------------------------------------------------------
           CMCS 2012 call for participation and short contributions
------------------------------------------------------------------------------

The 11th International Workshop on Coalgebraic Methods in Computer Science
                31 March - 1 April 2012, Tallinn, Estonia
                        co-located with ETAPS 2012
                           www.coalg.org/cmcs12

Aims and scope
--------------

In more than a decade of research, it has been established that a wide
variety of state-based dynamical systems, like transition systems,
automata (including weighted and probabilistic variants), Markov
chains, and game-based systems, can be treated uniformly as
coalgebras. Coalgebra has developed into a field of its own interest
presenting a deep mathematical foundation, a growing field of
applications, and interactions with various other fields such as
reactive and interactive system theory, object-oriented and concurrent
programming, formal system specification, modal and description
logics, artificial intelligence, dynamical systems, control systems,
category theory, algebra, analysis, etc. The aim of the CMCS workshop
series is to bring together researchers with a common interest in the
theory of coalgebras, their logics, and their applications.

The topics of the workshop include, but are not limited to:

   * the theory of coalgebras (including set theoretic and categorical
     approaches);
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FEJ Linton | 4 Feb 21:37

Re: discussing journals

[ For information only; no discussion please ]

Following up on what André Joyal wrote recently:

The Economist's latest issue (2012 Feb. 04, pp. 82-83) discusses Gowers'
blog and the resultant pledge/petition -- cf.:

http://www.economist.com/node/21545974  = and =
http://www.economist.com/node/21545974/comments .

Cheers, -- Fred

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Joyal, André | 5 Feb 22:43
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about to go up in smoke?

Information: 

http://www.forbes.com/sites/timworstall/2012/01/28/elseviers-publishing-model-might-be-about-to-go-up-in-smoke/

Cheers, ---André

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Andrej Bauer | 6 Feb 16:28
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Announcement: Fourth Workshop on Formal Topology (4WFTop) and Workshop on higher dimensional algebra, categories and types (HDACT)

  FOURTH WORKSHOP ON FORMAL TOPOLOGY (4WFTop)

                            June 15-19 2012

                                  and

HIGHER DIMENSIONAL ALGEBRA, CATEGORIES AND TYPES (HDACT)

                              June 20 2012

                          Ljubljana (Slovenia)

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

FOURTH WORKSHOP ON FORMAL TOPOLOGY (4WFTop)
===========================================

The workshop on formal topology is an international meeting dedicated
to formal topology and related topics, including constructive and
computable topology, point-free topology, and other generalizations of
topology.

This is the fourth of a series of successful meetings on the
development of Formal Topology and its connections with related
approaches. The first three have been held in Padua (1997), Venice
(2002), and Padua (2007).

IMPORTANT DATES

May 2      - deadline for abstract submissions
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Ohad Kammar | 8 Feb 21:29
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Pullbacks of a family of arrows

What is the standard name for the limit of the diagram consisting of a morphism
f : A --> C
and an I-indexed family of morphisms
g_i : B_i --> C
where I is a set?

This is a slight generalisation of pullbacks. I am really interested
in the terminology for the following notion:

A class of morphisms M is stable under such generalised pullbacks if,
for all such limiting cones
a : P --> A
b_i : P --> B_i
if g_i are all in M, then so is the morphism a : P --> A

(This is a slight generalisation of stability of M under pullbacks.)

Thanks,
Ohad.

--

-- 
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.

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