Octoberfest 2005
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
Abstracts
1. 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.

2. 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.

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.

4. Peter Freyd: Free Abelian Categories Revisited

5. 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.

6. 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.

7. 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.

8. 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*Sq. 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.

9. 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*).

10. 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 Z4. It is now suggested that Feynman diagrams contain arrows of a freely generated compact monoidal category.

11. 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 .

12. 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
cxh (m) : = hx-1(U cFx(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 cFx(Fm) is defined, one extends the notion of closure as cFx(Fm) = cFx(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 cr-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.

13. F.E.J. Linton: Triples vs. Theories as Yet another Yoneda Lemma
Abstract: see
http://home.att.net/~fej.math.wes/StreetFest/pdf/Linton FEJ.pdf

14. 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-1q-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-1F is a subset of f-1q-1F, 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.

15. 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.

16. 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 P2-construction for categories and 2-categories. This is joint work with Robert Dawson and Robert Paré.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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 TEX by TTH, version 3.35.
On 9 Oct 2005, 08:47.