Department of Mathematics & Statistics

585 King Edward

University of Ottawa

Schedule and Abstracts

** ROOMS NOTE: Saturday Morning talks & Registration in Montpetit 201; Saturday Afternoon
and Sunday talks are in Fauteux 351. Maps are on the Webpage. **

Saturday | Oct. 22 |

8:30-9:00 | Registration (Fee = $ 25) |

9:00-10:00 | Rick Jardine (Plenary Speaker) |

10:00-10:30 | Break |

Session A | |

10:30-11:00 | Jiri Rosicky |

11:00-11:30 | Gabor Lukacs |

11:30-12:00 | Claudio Hermida |

12:00-12:30 | Susan Niefield |

12:30-2:00 | Lunch (in Math Dept.) |

Session B | |

2:00-2:30 | Esfan Haghverdi |

2:30-3:00 | Brian Redmond |

3:00-3:30 | Sergey Slavnov |

3:30-4:00 | Jeff Egger |

4:00-4:30 | Break |

Session C | |

4:30-5:00 | Nicola Gambino |

5:00-5:30 | Guy Beaulieu |

5:30-6:00 | Dorette Pronk |

6:00-6:15 | |

Sunday | Oct. 23 |

Session D | |

9:00-9:30 | F. W. Lawvere |

9:30-10:00 | J. Lambek |

10:00-10:30 | M. Weber |

10:30-11:00 | Break |

Session E | |

11:00-11:30 | Peter Freyd |

11:30-12:00 | Robin Cockett |

12:00-12:30 | Walter Tholen |

12:30-1:30 | Lunch (in Math Dept.) |

Session F | |

1:30-2:00 | Jon Funk |

2:00-2:30 | Larry Stout |

- Guy Beaulieu: Adding Probabilistic Capabilities to Models of
Nondeterminism.
Abstract: I shall motivate and construct the Lawvere theory of mixed
choice, which combines nondeterministic and probabilistic operators.
Interestingly, the monad associated to the Lawvere theory of mixed
choice is not the composition of the monads associated to the Lawvere
theories of nondeterministic choice and probabilistic choice.
However, we prove a factorization theorem which states its relation
to the Eilenberg-Moore adjunctions capturing nondeterministic and
probabilistic choice. Finally, we discuss the intricacies of adding
probabilistic choice capabilities to a non-free model of
nondeterministic choice.
- Robin Cockett: The partial lambda calculus
(Joint work with Pieter Hofstra)
It is a classic result for the lambda-calculus that a
lambda-algebra gives rise to a C-monoid and whence a cartesian
closed category. The purpose of the talk is to show how this result
generalizes to the partial case.
Taking the place of a cartesian closed category is a cartesian closed
restriction category which is a formal category of partial maps with
partial products and exponentials. The role of a lambda-algebra is
taken by a partial combinatory algebra, sitting in an arbitrary
restriction category, which must interpret the partial
lambda calculus.
We are now looking for models of these structures! A useful observation,
in this regard, is that the D-infinity construction works almost
verbatim (over DCPOs with partiality given by open inclusions) to
produce extensional partial lambda-models.
- Jeff Egger: More about adherence spaces.
Abstract: In this talk, I will answer a question (posed by me at the
end of my talk at Union College in November 2003) about when a
category of adherence spaces is *-autonomous.
- Peter Freyd: Free Abelian Categories Revisited
- Jon Funk: Hyperpure geometric morphisms
(Joint work with Marta Bunge.)
Abstract: A geometric morphism is said to be hyperpure [3] if its
inverse image functor reflects definable suprema of definable
subobjects (definable subobject and morphism are taken in the sense
of Barr and Pare [1]). This notion originates in topology with
Michael [4]. For instance, a map of locally connected spaces is
hyperpure just when its inverse image functor takes connected opens
to connected ones. In the locally connected case, these maps have
been called pure: such a map is the `epi' factor of the so-called
comprehensive factorization of a geometric morphism with locally
connected domain [2]. Our main result [3] is that an arbitrary
geometric morphism may be uniquely factored into a hyperpure one
followed by a complete spread. Comprehensive factorization is an
instance of this new factorization. The 0-dimensional reflection of a
topos is also a special case of the new factorization. It makes sense
therefore to say that a topos EE is (at most) 1-dimensional without
boundary iff every hyperpure monomorphism U >--> V (meaning that the
geometric morphism EE/U --> EE/V is hyperpure) in EE is an
isomorphism.
[1] M. Barr and R. Pare, Molecular toposes. JPAA. 1980. [2] M. Bunge
and J. Funk, Spreads and the symmetric topos, JPAA, 1996. [3] M.
Bunge and J. Funk, Quasicomponents in topos theory: the hyperpure,
complete spread factorization. Submitted for publication. 2005. [4]
E. Michael, Completing a spread (in the sense of Fox) without local
connectedness, Indag. Math. 1963.
- Nicola Gambino: Pseudo-distributive laws for Kleisli structures
Abstract: It is well-known that the category of presheaves over a
monoidal category carries a canonical monoidal structure given by
Day's convolution tensor product.
In my talk, I will present an analysis of situations of this form via
the notions of a Kleisli structure and of a pseudo-distributive law.
This extends work of Kelly, Lack, Marmolejo, and Tanaka.
Applications will be given to Joyal's theory of species of
structures, Baez and Dolan's generalised operads, Ehrhard and
Reigner's differential lambda-calculus, and Winskel's theory of
concurrency.
- E. Haghverdi: Towards a Typed Geometry of Interaction
(joint work with P.J. Scott)
Girard's Geometry of Interaction (GoI) develops a mathematical
framework for modelling the dynamics of cut-elimination. We introduce
a typed version of GoI, called Multiobject GoI (MGoI) for
multiplicative linear logic without units in categories which include
previous (untyped) GoI models, as well as models not possible in the
original untyped version. The development of MGoI depends on a new
theory of partial traces and trace classes, as well as an abstract
notion of orthogonality (related to work of Hyland and Schalk.) We
develop Girard's original theory of types, data and algorithms in our
setting, and show his execution formula to be an invariant of Cut
Elimination. We prove Soundness and Completeness Theorems for the
MGoI interpretation in partially traced categories with an
orthogonality. Moreover, as an application of our MGoI
interpretation, we prove a completeness theorem for the original
untyped GoI interpretation of MLL in a traced unique decomposition
category.
- Claudio Hermida: Descent on 2-fibrations and strongly 2-regular
2-categories
We consider pseudo-descent in the context of 2-fibrations. A
2-category of descent data is associated to a 3-truncated simplicial
object in the base 2-category. A morphism q in the base induces (via
comma-objects and pullbacks) an internal category whose truncated
nerve allows the definition of the 2-category of descent data for q .
When the 2-fibration admits direct images, we provide the analogous
of the Beck-Benabou-Roubaud theorem, identifying the 2-category of
descent data with that of pseudo-algebras for the pseudo-monad
q*S
_{q}. We introduce a notion of 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod: Fib (R) --> R admits direct images. In this context, we show that essentially-surjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem. - R. Jardine: Homotopy classification of gerbes
Suppose that E is a sheaf of sets. An E-gerbe is a presheaf (or
sheaf, or stack) of groupoids G with a choice of isomorphism of E
with the sheaf of path components of G. Weak equivalences of E-gerbes
can be identified with path components of a category of cocycles over
E taking values in the 2-groupoid of sheaves of groups, their
isomorphisms and homotopies. This result specializes to an
identification of E-gerbes locally equivalent to a fixed object G and
path components of cocycles taking values in the 2-groupoid G
_{*}of sheaves of groups which arise as fundamental groups of G. Gerbes locally equivalent to G with a fixed choice of band arise as points in a homotopy fibre for a canonical map B(G_{*}) --> B(OutG_{*}). - J. Lambek: Some ideas for applying compact monoidal categories to
Linguistics and Physics.
A useful idea in natural language processing has been the assignment
of types of words. In my recent approach the types have been elements
of a free pregroup. To describe grammatical derivations one may use
arrows of a compact monoidal category. One recent idea for describing
the particles of the Standard Model in physics has been to attach to
each of them an element of Z
_{4}. It is now suggested that Feynman diagrams contain arrows of a freely generated compact monoidal category. - F. William Lawvere:
Presenting the algebra of negations: unity & identity of adjointly
opposite operations
Three-valued logic is equivalent to two-valued, but enjoys the
advantage that unary operations on it are co-adequate. To
computationally exploit the latter observation requires presentation
of that monoid of unary operations; but even order-preserving
operations suffice, reducing 27 to 10. A microanalysis of the 10 can
be gained by regarding 3 as parameterizing order-reversing maps from
V to V, where V is the two-element ordered set. The system N of such
"negations" makes sense for any self-total symmetric monoidal closed
category V; in the case V=sets it has been studied by Rosebrugh &
Wood in their characterization V of sets, and subsumes the remarkable
near-ubiquity of the very special "toposes generated by codiscrete
objects". The endofunctors of the "presheaf" category N obtained by
composing the idempotents arising in canonical adjoint cylinders are
apparently co-adequate for N-valued logic, at least for N=3 .
- Gabor Lukacs: Applications of lifted closure operators in topology
and topological algebra
Let X and A be finitely-complete categories with proper (E,M) and
(F,N) factorization systems, respectively. If F --| U: A -->X is an adjunction
with unit h, and c is a closure operator on A
with respect to N, then one defines the F-initial lift of c as
c
_{x}^{h}(m) : = h_{x}^{-1}(U c_{Fx}(Fm)) for every object x in X, and subobject m in sub(x). The morphism Fm need not belong to N, so in order to ensure that c_{Fx}(Fm) is defined, one extends the notion of closure as c_{Fx}(Fm) = c_{Fx}(Im(Fm)). In this talk, we discuss the special case of this construction where A is a reflective subcategory of X with reflector F=R and reflection h = r. We provide a characterization of c^{r}-compact objects in X under certain conditions. The following applications are presented: (A) X=Top, A=Tych, and R= t the Tychonoff reflection; (B) X=Grp(Top)(topological groups and their continuous homomorphisms), A=Grp(HComp) (compact Hausdorff groups and continuous homomorphisms), and R=b the Bohr-compactification; (C) X=T^{*}A the category of topological *-algebras and their continuous *-homomorphisms, A the full subcategory of pro-C*-algebras (i.e., limits of C*-algebras in T^{*}A), and R the functor assigning to each A in T^{*}A its enveloping pro-C*-algebra. - F.E.J. Linton: Triples vs. Theories as Yet another Yoneda Lemma
Abstract: see
`http://home.att.net/~fej.math.wes/StreetFest/pdf/Linton` - Susan Niefield: Exponentiability in Lax Slices of Top
Let B be a topological space, and let Top\nearrow B denote the
category whose objects are continuous maps p:X ® B and
morphisms are triangles which commute up to the specialization order <=
on B, i.e., a morphism (X,p)--> (Y,q) is a continuous map f:X-->Y such that
px<=qfx, for all x in X, or equivalently, p
^{-1}(U) is a subset of f^{-1}q^{-1}(U), for all U open in B. One can show that Top\nearrow B has products which are preserved by the forgetful functor to Top if and only if (B, < =) is a meet-semilattice and meet:BxB- > B is continuous. The category Top\nearrow B can be used to study certain B-indexed diagrams of open subsets of topological spaces. For example, if 2=0,1 is the Sierpinski space with 1 open but not 0, then Top\nearrow 2 is isomorphic to the category whose objects are pairs (X,U) of spaces with U open in X, and morphisms (X,U)--> (Y,V) are continuous maps f:X --> Y such that f(U) is a subset of V. Similarly, Top\searrow B is defined by requiring morphisms to commute up to >= (or equivalently, p^{-1}F is a subset of f^{-1}q^{-1}F, for all F closed in B), and Top\searrow B has products which are preserved by the forgetful functor if and only if (B,<=) is a join-semilattice and join:BxB --> B is continuous. Moreover, Top\searrow B can be used to study certain B-indexed diagrams of closed subsets of topological spaces. For example, Top\searrow 2 is isomorphic to the category whose objects are pairs (X,F) with F closed in X, and morphisms (X,F) --> (Y,G) are continuous maps f:X --> Y such that f(F) is a subset of G. In this talk, we consider exponentiable objects in the lax slice category Top\nearrow B (respectively, Top\searrow B) when B is a complete Heyting (respectively, co-Heyting) algebra with the Alexandrov topology. - Cyrus Nourani: Fragment
Consistent Algebraic Models
Preliminaries Abstract Fragment Consistency Models are presented with
new techniques for creating generic models. Infinitary positive
language categories are defined and infintary complements to Robinson
consistency from the author's preceding paper are gleaned to present
new positive omitting types techniques to infinitary positive
fragment higher stratified consistency. Further classic
model-theoretic consequences are presented. Except otherwise stated
all definitions, theorems and propositions are new here or due to the
author.
- Dorette Pronk: The Pathology of Double Categories
Abstract: We will present the path construction for double categories
and show how it sheds a new light on the construction of universal
(op)lax (normal) morphisms (both for double categories and
bicategories), as well as the the P
_{2}-construction for categories and 2-categories. This is joint work with Robert Dawson and Robert Paré. - Brian Redmond: Categorical Models of Soft Linear Logic and P-time Computation

Abstract: Certain variants of Linear Logic can lay claim to being complete for P-time computation, e.g. Bounded Linear Logic, Light Linear Logic, Soft Linear Logic, etc. In general these logics rely on different logical principles which seem difficult to unify. It is hoped that their categorical interpretation might lead to a common framework. In this talk, we shall discuss a categorical interpretation of Soft Linear Logic. After considering a few examples of models, we construct a new category which leads to an interesting family of models. In particular, we get a realizability semantics for SLL. We end by noting a surprising connection between this category and P. Baillot's category SCOH, of stratified coherence spaces for LLL. - Jiri Rosicky: Homotopy varieties
Abstract: Given an algebraic theory T, a homotopy T-algebra is a
functor from T to the category of simplicial sets which preserves
finite products up to a weak equivalence. All homotopy T-algebras
form a homotopy variety. One can even consider homotopy varieties
given by simplicial algebraic theories, i.e., by small simplicial
categories with finite products (a homotopy algebra is then a
simplicial functor). We will give a characterization of homotopy
varieties analogous to the characterization of varieties. It yields a
rigidification theorem for simplicial algebraic theories saying that
every homotopy algebra is weakly equivalent to a strict algebra; this
theorem has been recently proved by B. Badzioch and J. E. Bergner in
the special case of algebraic theories. There are similar results for
homotopy limit theories leading to homotopy locally presentable
categories which have been recently considered by C. Simpson, J.
Lurie, B. Toen and G. Vezzosi. There is also a rigidification theorem
here.
- Sergey Slavnov: Is there any geometry in Geometry of Interaction?
Geometry of Interaction (GoI) was defined by J.-Y. Girard as a
special kind of model for linear logic. Later S. Abramsky gave his
own, extremely elegant, categorical definition of GoI, in some sense
generalizing Girard's construction. It is well-known that linear
logic can be modeled in any *-autonomous category, by interpreting
proofs as (di-)natural morphisms. In Geometry of Interaction one
requires further that normalization of proofs (the "dynamics") be
reflected in the model in some sensible way. For example, in
Abramsky's abstract version of GoI normalization is modeled by the
categorical trace. Unfortunately this is possible only if the
*-autonomous structure of the model category is degenerate, hence
the model itself is degenerate. We argue that there is an essential
difference between Abramsky's and Girard's formulations, and that
there is a correct generalization of Girard's construction, which
produces non-degenerate *-autonomous categories. We note further that
this construction fits perfectly well into the basic framework of
Hamiltonian dynamics and the underlying symplectic geometry of phase
spaces. Thus we consider a category, whose objects are phase spaces,
and whose morphisms are canonical transformations (canonical
relations). We note that the category has a non-degenerate
*-autonomous structure and discuss its logical and geometrical
meaning.
- Larry Stout: A View of the Continuum Using Categories with Inclusion
Structures
Abstract: At the meeting of the Association for Symbolic Logic in
Urbana in 2000 Mac Lane proposed consideration of inclusion
structures on categories (perhaps in an attempt to attract set
theorists to categorical foundations- indeed, models of set theory
can be characterized using categories with designated inclusions).
Inspired by that talk I've specified one possible definition for an
inclusion structure on a category. This paper gives a definition and
examples to show the utility of the concept. The main example I
consider comes from a categorical approach to the geometry of the
continuum built on closed intervals as objects. The resulting
category has few desirable structures, but when equipped with an
inclusion structure captures all of the important characteristics of
the continuum.
- Walter Tholen: Characterization of subcategories closed under
extensions (Joint work with George Janelidze, University of Cape
Town)
Abstract: In a pointed category with kernels and cokernels, we
characterize torsion-free classes in terms of their closure under
extensions. We obtain a corresponding characterization of torsion
classes by formal dualization. Critical counter-examples are also
presented.
- Mark Weber: 2-toposes and higher categories
Abstract: In this talk a definition of *2-topos*, inspired by Bill
Lawvere's idea that the category of sets is a generalised object of
truth values, and the Cosmos theory of Ross Street, together with the
appropriate 2-categorical analogue of *natural numbers object* will
be presented. Applications of these notions to the globular approach
to higher category theory will be discussed.

File translated from T

On 9 Oct 2005, 08:47.