1 Jan 15:59 2011

Does this topology have a name?

Let A be a model of a finitary equational theory and let X be the set of
congruences on A.  For a,b in A, let M(a,b) = {E} such that E is a
congruence on A and aEb.  Does this topology have a name?  It turns out
that this topology is coherent which means, among other things, that if we
make the M(a,b) clopen, the result is a Stone space.

Obviously in a ring, we could instead use the set of ideals, but aside
from the fact that that will include non-prime ideals, the topology is the
opposite of the Zariski topology.

Michael

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


1 Jan 19:53 2011

Re: Does this topology have a name?


On Jan 1, 2011, at 6:59 AM, Michael Barr wrote:

> Let A be a model of a finitary equational theory and let X be the set of
> congruences on A.  For a,b in A, let M(a,b) = {E} such that E is a
> congruence on A and aEb.  Does this topology have a name?  It turns out
> that this topology is coherent which means, among other things, that if we
> make the M(a,b) clopen, the result is a Stone space.

Consider the powerset space P(A x A) = 2^(A x A).  The product topology
makes it a Stone space.  This is elementary.

Now, the space X of congruences is defined by logical formulae with
only universal quantifiers and atomic formulae xEy for variables
ranging over A.  That makes X a CLOSED subspace of P(A x A).

This is so easy, it hardly needs a name.  And it works even if A has
infinitely many operations.  That there is an equational theory in the
background seems neither here nor there to get a Stone space of congruences.

HAPPY NEW YEAR!

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


1 Jan 23:15 2011

Re: Does this topology have a name?

Yes, the fact that when these sets are taken as clopens gives a Stone
space is easy.  But I want to know what to call the weaker topology in
which you take these sets as a basis of opens.

Happy New Year to you!

Michael

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


2 Jan 17:20 2011

LU Special Issue on Categorical Logic: deadline reminder & extension

hi,
Due to some emailing mishap the email below wasn't circulated, we
take the opportunity to extend the deadline to 31st January 2011.

Happy Holidays and a Great New Year!

Valeria and Andre

Dear Colleagues,
the earlier announced deadline January 1, 2011 for submissions to a
special issue of Logica Universalis on categorical logic is now
approaching; details of this call are found on the journal's website:
www.logica-universalis.org.
very best wishes,
Andrei Rodin

--

--
Valeria de Paiva
http://www.cs.bham.ac.uk/~vdp/

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


2 Jan 19:30 2011

Re; Does this topology have a name?

I wrote to Barr as follows:

> From: Dana Scott <dana.scott <at> cs.cmu.edu>
> Date: January 1, 2011 2:57:30 PM PST
> To: Michael Barr <barr <at> math.mcgill.ca>
> Subject: Re: categories: Does this topology have a name?
>
>
> On Jan 1, 2011, at 2:15 PM, Michael Barr wrote:
>
>> Yes, the fact that when these sets are taken as clopens
>> gives a Stone space is easy.  But I want to know what to
>> call the weaker topology in which you take these sets as
>> a basis of opens.
>
> Ah, I had thought you meant the clopen case.  The weaker
> topology is (unfortunately) called the Scott topology,
> which can be given to any algebraic lattice.  The
> congruences form an algebraic lattice inasmuch as they
> are closed under arbitrary intersections and directed
> unions.  (Yes?)  Details are in the book: Continuous
> Lattices and Domains by Gierz/Hofmann/Keimel/Lawson/
> Mislove/Scott.

A little more detail: The opens in the lattice of congruences
are determined by the "compacts" of this algebraic
lattice.  These are the finitely generated congruences.
If F is one such, then the open it determines is
{E | F subset E}.  They form a basis for the "Scott"
topology.  A subbasis is given by the sets {E | aEb}


3 Jan 00:26 2011

source, sinks, and ?

A family of morphisms { x_i --> y }_{i \in I} in some category, all
with the same codomain, is called a "sink" or a "cocone".  A family {
x --> y_j }_{j \in J} all with the same domain is called a "source" or
a "cone".  Is there a name for a family of the form { x_i --> y_j }_{i
\in I, j \in J} ?  A cylinder?  Or a frustrum (since I \neq J)?

Mike

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


3 Jan 01:40 2011

Re: source, sinks, and ?

On Sun, 02 Jan 2011 07:17:17 PM EST Michael Shulman <mshulman <at> ucsd.edu>

> ... Is there a name for a family of the form { x_i --> y_j }_{i
> \in I, j \in J} ?  A cylinder?  Or a frustrum (since I \neq J)? ...

Looks more like a "mish-mash" to me  . Cheers, --  Fred

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


3 Jan 05:19 2011

Kan's Ex functor for the contravariant homotopy structure

I've been thinking about the following question, and I was wondering
if anyone else has given it some thought:

Is there a good notion of an Sd/Ex adjunction for sSet/S equipped with
the contravariant model structure (cofibrations are monomorphisms and
fibrant objects are right fibrations over S) for an arbitrary
simplicial set S? (Note: This is in the _unmarked_ case.)

It seems to me that any sort of naive way of doing this (for instance,
by pulling back the results in sSet (that is, given an object p:X->S
of sSet/S, let

Ex_S(p):=S\times_{Ex(S)} Ex(X) -> S

with morphisms determined by the universal property)) is doomed to
fail, since it does not incorporate the asymmetry of the model
structure (that is, if that worked, it would also work for the
covariant model structure, which seems like it shouldn't be true).

One problem with trying to mimic the classical argument is that the
classical/Quillen/Kan homotopy structure (this comprises the data of
the model structure on sSet and all of its relativizations sSet/S for
every simplicial set S (see Cisinski's book _Les Prefaisceaux comme
modeles des types d'homotopie_ ch 1.3 for a precise definition, as
well as some relevant results)) has the property of _completeness_,
which is essentially the property that the weak equivalences of sSet/S
are exactly the morphisms that map to weak equivalences under the
canonical projection functor sSet/S -> sSet.  Since the contravariant
homotopy structure does not have this property, it seems imprudent to
expect to be able to pull back results from "deeper" bases naively.


3 Jan 12:43 2011

European Workshop on Computational Effects


EUROPEAN WORKSHOP ON COMPUTATIONAL EFFECTS

Thursday 17th and Friday 18th March 2011, Ljubljana, Slovenia.
http://ewce.fmf.uni-lj.si/

ANNOUNCEMENT AND CALL FOR CONTRIBUTED TALKS

The aim of the workshop is to bring together researchers investigating
computational effects from a variety of different angles: programming
languages, type theory, operational semantics, universal algebra,
category theory, denotational semantics, etc.

CONTRIBUTED TALKS

A limited number of slots are available for contributed talks. Please
submit a title and short text abstract by email to

Alex.Simpson <at> ed.ac.uk

acceptance will be by Monday 7th February 2011.

INVITED SPEAKERS

Nick Benton, Microsoft Research, Cambridge
Andrzej Filinski, Copenhagen University
Paul Blain Levy, University of Birmingham
Paul-Andre Mellies, Paris 7


3 Jan 23:40 2011

Re: source, sinks, and ?

Dear Mike,

In fact, we can call it a matrix (of (I x J)- type)  with coefficients
in the category. When the category is a category of modules, we have
that way a natural generalization of classical matrices.

Cheers,
Albert

Michael Shulman <mshulman <at> ucsd.edu> a écrit :

> A family of morphisms { x_i --> y }_{i \in I} in some category, all
> with the same codomain, is called a "sink" or a "cocone".  A family {
> x --> y_j }_{j \in J} all with the same domain is called a "source" or
> a "cone".  Is there a name for a family of the form { x_i --> y_j }_{i
> \in I, j \in J} ?  A cylinder?  Or a frustrum (since I \neq J)?
>
> Mike

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



Gmane