Seminars of the
CENTRE de RECHERCHE en THEORIE des CATEGORIES
CATEGORY THEORY RESEARCH CENTER
C ---------> R
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T ---------> C
30 Jan 2001
2:30 - 5:30 Eduardo Dubuc
"Localic Galois Theory I and II"
(There will be a half hour break, 3:30 - 4:00 for
the usual coffee and cookies, and conversation.)
13 Feb 2001
2:30 - 4:00 R.A.G. Seely
A Logic of a Linear Functor
20 Feb 2001
2:30 - 4:00 Eduardo Dubuc (Universidad de Buenos Aires)
Localic Galois Theory III
(This talk will be self-contained
and may be attended by folks who
missed the first two parts)
27 Feb 2001
2:30 - 4:00 Jonathan Funk
Unramified cosheaf spaces.
(Report on joint work with Ed Tymchatyn)
Tuesday, 6 March 2001
2:30 - 4:00 Robert Paré
Kan extensions for double categories
7 Mar 2001
2:30 - 4:00 Jeff Egger
Some little known folklore about Zorn's Lemma
This talk, on a WEDNESDAY, will take place in BH 1120 instead of the
usual 920.
Abstract:
This will be the first of a series of talks investigating the Axiom of
Choice for Finite Sets [KC] in the context of topos theory.
Classically, KC follows from the Linear Ordering Principle, which is a
consequence of the Prime Ideal Theorem, which is usually proven using
Zorn's Lemma. So it seems appropriate to begin with a reminder of
what is known about Zorn's Lemma in toposes, and in particular, how it
differs from the Axiom of Choice.
Wednesday, 14 March 2001
2:45 - 4:15 Richard Squire
Internal maximality in non-localic toposes
Abstract:
In general, partial orders which are maximal in the external sense
are not maximal in the internal sense. We shall show that for a
general notion of sketch toposes external will coincide with internal.
Location for the Wednesday talks: BH1120
(Tuesday talks remain in BH920)
27 Mar 2001
2:30 - 4:00 Bob Coecke
From Birkhoff - von Neumann to Eilenberg - Mac Lane:
Categories for the digesting physicist
(Abstract on web page)
Tuesday, 3 April 2001
2:30 - 4:00 Stephen Watson
Transfinite Recursion without the Details
Abstract:
We describe how many difficult transfinite recursions can be defined
by removing the quantifiers of a mathematical instruction by
relativizing them according to a schedule to elementary submodels and
finite approximations thereof. Our examples come from set-theoretic
topology but may be broadly applicable.
Tuesday, 10 April 2001
2:30 - 4:00 Richard Wood
Decomposing Regularity
Abstract:
Many important classes of categories are specified by certain types of
colimits, certain types of limits, and exactness conditions relating
these. If the colimits are given by a KZ-doctrine R and the limits by
a co-KZ-doctrine L then it makes sense to enquire about the existence
of a distributive law LR--->RL in the sense of Beck. (Essentially,
there is at most one such law in this context.) Given such a law, an
algebra for the composite doctrine RL is a category C with
colimits as prescribed by R,
limits as prescribed by L,
specification of R-colimits, RC--->C, L-limit-preserving.
Examples will be given. The talk will report on joint work with
Claudia Centazzo directed towards the problem of solving D=RL, for R,
where D is the doctrine for regular categories and L is the doctrine
for categories with finite limits.
Wednesday, 11 April 2001
2:30 - 4:00 Richard Squire
Topos-theoretic characterization of finite-valued presheaves
Location: BH920
(Abstract on web page)
Tuesday, 17 April 2001
2:30 - 4:00 Bob Coecke
From Birkhoff - von Neumann to Eilenberg - Mac Lane:
Categories for the digesting physicist II
(Abstract on web page)
Wednesday, 18 April 2001
2:30 - 4:00 Jeff Egger
The L-prime Ideal Theorem, and its Corollaries
Abstract:
The completeness theorem for coherent propositional logic, alias
the Prime Ideal Theorem (PIT) for distributive lattices, is true
in certain non-boolean toposes. But it is only strong enough to
prove the Order Extension Principle (OEP) for decidable objects.
We introduce an extension of coherent propositional logic whose
completeness theorem is stronger than (PIT). In particular, it
is strong enough to conclude (OEP) for arbitrary objects.
This is the second in a series of talks concerning the axiom of
choice for (Kuratowski-)finite sets and ``related issues''.
Location: BH920
Wednesday, 25 April 2001
2:30 - 4:00 Richard Squire
Topos-theoretic characterization of finite-valued
presheaves II - "Issues" and a counter-example
(Location: BH 920)
Wednesday, 2 May 2001
2:30 - 4:00 Jeff Egger
Subsingletons and genuine subsingletons
(Location: BH 920)
Abstract: We introduce the notion of `genuine' subsingleton, which is
relevant both to my investigation of Zorn's Lemma and my study of
Vermeulen-finitary algebraic structures. Proofs from my last two
talks will be improved and/or clarified, and at least one converse
will be added.
Tuesday, 11 September 2001
2:30 - 3:30 Bob Coecke
QUESTIONS on physical logicality
(vs. constructivism and resource sensitive provability)
4:00 - 5:00 Boris Ischi
Property lattices for separated quantum systems
Tuesday, 18 September 2001
2:30 - 4:00 Prakash Panangaden
Discrete Quantum Causal Dynamics
(Joint work with Rick Blute and Ivan T Ivanov)
(Abstract on web page)
Tuesday, 25 September 2001
2:30 - 4:00 Prakash Panangaden
Discrete Quantum Causal Dynamics II
(Joint work with Rick Blute and Ivan T Ivanov)
(Abstract on web page)
Tuesday, 2 October 2001
2:30 - 4:00 M Barr
Flat modules in localic toposes
Tuesday, 9 October 2001 CANCELLED
2:30 - 4:00 Alexander Nenashev (U Sask)
The theory of quadratic forms in the
framework of exact or triangulated
categories with duality.
Abstract. Such classical notions as quadratic forms, hyper- and
metabolic forms, and Witt groups can be introduced and studied over
exact or triangulated categories with duality. There are classical
invariants of quadratic forms over a field of char different from 2.
In low degrees these are the dimension index, the reduced
discriminant, and the Hasse-Witt invariant. We discuss a possibility
of introducing analogous invariants for quadratic forms over exact
categories with duality, with values in suitable subquotients of the
higher K-groups. Over triangulated categories with duality, Witt
theory has been developed recently by P. Balmer, with interesting
applications to the Witt groups of algebraic varieties.
Tuesday, 6 November 2001
2:30 - 4:00 Rezaei Siamak
From categorial to process grammars
Tuesday, 13 November 2001 CANCELLED
3:00 - 4:30 Claudia Casadio
Word order and scope in pregroup grammar
Tuesday, 4 December 2001
2:30 - 3:30 Bob Coecke
Physical realization of the traced monoidal category
of Finite dimensional vector spaces.
(Joint work with Samson Abramsky)
Abstract
Exploiting the phenomenon of quantum entanglement (that will be
explained) we can operationally realize a setup that mimics the trace
construction for the category of finite dimensional vector spaces and
as such, via the geometry of interaction construction, the
multiplicative fragment of linear logic.
This also exhibits the operational difference between the traced
monoidal category of sets and partial functions (or relations) either
with disjoint union or cartesian product as tensor.
4:00 - 5:00 Claudia Casadio
Word order and scope in pregroup grammar
Tuesday, 11 December 2001
2:30 - 3:30 Luigi Santocanale
Fixed Point Logics and Circular Proofs
ABSTRACT: Fixed point logics and $\mu$-calculi are obtained
from previously existing logical or algebraic frameworks by
the addition of least and greatest fixed point operators.
The propositional modal $\mu$-calculus or modal $\mu$-logic,
useful for model checking, arises from modal logic exactly
in this way.
A proposition-term in a $\mu$-calculus has a circular
structure: it is possible to travel along subformulas and
come back to the starting point using regenerations of fixed
point. The circularity is inherited by proof-terms.
Normally cut-free proofs in sequent calculi are finite
because premiss sequents are always strictly smaller than
conclusions. However, in settings where the propositions
themselves can be circular, there exists the possibility of
having circular or infinite proofs as well. Remarkably, in
the theories of fixed points, these kind of proofs happen to
be the most useful. Thus, the mathematics suggests that we
are indeed allowed to make circular reasonings.
In this talk I'll explain this apparent paradox, using the
concept of initial algebra (and final coalgebra) of a
functor. I'll point out the sense for which initial algebras
of functors are a generalization of least fixed points and
the relationship to inductively constructed sets (such as
lists, finite tress, etc.).
I will interpret circular proofs as sort of functions (more
precisely, arrows of a category) having as domain an
inductively constructed set. Recall that recursively
defined functions are uniquely determined by their defining
system of equations. Similarly it is possible to transform a
circular proof in a system of equations and then prove that
this system admit a unique solution.
NOTE: Luigi will also be talking at UQAM:
Pavillon Président-Kennedy, local PK-4323
Friday December 14
Jouer avec l'induction et la coinduction: les jeux de
parité.
Un jeu de parité est joué sur un graphe fini de positions et
de mouvements. Dans ce graphe on peut bien avoir des cycles,
et pourtant on définit l'ensemble des chemins inifinis
gagnants pour un des deux joueurs à l'aide de la ``condition
de parité''. Cette condition est bien connue dans la théorie
des automates qui reconnaissent les objects infinis. En
effet tout automate est équivalent à un automate qui utilise
la condition de parité. Nous allons définir cette condition
et lui donner une interprétation algébrique à l'aide des
notions d'algèbre initiale et coalgèbre finale d'un
foncteur, c-à-d., à l'aide des formulations catégorielles de
l'induction et de la coinduction. Nous allons associer une
expression algébrique à chaque jeu de parité et montrer
qu'elle dénote (dans la catégorie des ensembles) l'ensemble
des stratégies gagnantes pour un des deux joueurs. Nous
montrons aussi que les expressions associés aux jeux sont
équivalentes aux expressions qu'on peut engendrer par les
operations de produit fini, de coproduit fini, d'algèbre
initiale et coalgèbre finale, c.à-d aux µ-termes
catégoriels. Nous obtenons cette façon une caractérisation
explicite de tous les foncteurs qu'on peut définir par des
µ-termes dans la catégorie des ensembles.
(Apologies for any effects of bad character coding!)
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COFFEE: Coffee and cookies will be available after the talk in the lounge.
PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY
===================================================
(Any comments, suggestions to rags@math.mcgill.ca)
Seminar listings are also on the triples WWW page
http://www.math.mcgill.ca/triples
===================================================
COFFEE: Coffee and cookies will be available after the talk in the
lounge.
PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY
===================================================
(Any comments, suggestions to rags@math.mcgill.ca)
Seminar listings are also on the triples WWW page
http://www.math.mcgill.ca/triples