Tuesday, July 21, 2009
Geoff Cruttwell, A Unified Framework for Generalized Multicategories
Abstract: Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the ``lax algebras'' or ``Kleisli monoids'' relative to a ``monad'' on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.

This is joint work with Mike Shulman, and is an expanded version of my CT'09 talk. It will also be my last talk before leaving for Calgary.

Tuesday, September 1, 2009
Jeffrey Morton, Groupoidification and 2-Linearization in Quantum Physics
Abstract: Quantum mechanics and quantum field theory are usually presented in the category of Hilbert spaces. In this talk, I will describe how the operations of degroupoidification and 2-linearization described in the first talk appear in some examples from these settings, as well as discussing other categorical structures involved. In particular I will describe how a simple quantum mechanical system can be related by groupoidification to the "combinatorial species" of Joyal; and how a generalization of Topological Quantum Field Theory (TQFT) can be constructed in terms of 2-linearization, including a discussion of the higher-categorical structures (double bicategories) used to describe a generalized form of cobordism between manifolds, and the relation to some work by Weinstein on smooth stacks.

Tuesday, September 15, 2009
Richard Wood, Monads as Extension Systems (joint with Francisco Marmolejo)

Tuesday, September 22, 2009
Margaret Beattie, Cocycle deformations for Hopf algebras with a coalgebra projection
Abstract: Let H be a sub-Hopf algebra of a bialgebra A such that there is an H-bilinear coalgebra projections from A to H which splits the inclusion. The coinvariants of this projection are then a pre-bialgebra; if the projection is a bialgebra map, the coinvariants are a bialgebra in the Yetter-Drinfeld category. We study twistings of A by a cocycle and how these correspond to twistings of the associated pre-bialgebra where here the notion of a cocycle must be suitably defined.

Tuesday, October 20, 2009
Octavio Malherbe, Presheaf models of quantum lambda calculus
Abstract: I will attempt to show how to build presheaf models of quantum lambda calculus in the sense of Valiron and Selinger.

Tuesday, November 3, 2009
Richard Wood, Completely Distributive and Totally Distributive Categories (Joint with Bob Rosebrugh)
Abstract: A locally small category is said to be totally cocomplete if its Yoneda functor Y has a left adjoint, X. In a paper characterizing the category of sets among totally cocomplete (total) categories we introduced the terminology {\em totally distributive} for those total categories for which X has a left adjoint, W. We showed that small powers of the category of sets are totally distributive. (In fact we characterized these as the totally distributive categories for which the inverter of the canonical W==>Y is dense and Kan.)

It is also known that the category of sheaves on a CCD lattice is totally distributive. In this talk we show that a further supply of totally distributive categories is provided by categories of distributors between small {\em taxons}. A taxon is defined like a category except that the existence of identities is dropped in favour of requiring merely that composition be a colimit. There is a 2-category of taxons and a 2-functor i:cat--->tax that interprets a category as a taxon. We show that for any small taxon A, tax(A^op,i(set)) is a totally distributive category. It is temptintg to conjecture that all totally distributive categories arise in this way.

Tuesday, November 10, 2009
Bob Rosebrugh, EASIK: categorical database design and manipulation
Abstract: Finite-limit, finite-sum sketches (aka EA sketches) are the right syntactic structure for modelling databases and their "views". Mike Johnson, RJ Wood and I have explored this observation for several years. In particular, the ability to express constraints and determine updatability of views both extends and considerably enhances those of the standard Entity-Relationship (ER) data model. We will present a short overview of these ideas.

Using Java, several Mount Allison students and I have written an application that provides a user-friendly graphical design environment for EA sketches, supports exporting a design to a database schema in SQL (the standard relational database language), and recently Brett Giles has extended this to the Web-friendly XML. EASIK also allows data entry and manipulation. The most recent version of the application will be demonstrated.

Tuesday, November 24, 2009
Bob Paré, The quantale of a category
Abstract: The set of subsets of arrows of a small category has a quantale structure induced by composition. We study the interplay between quantale properties and category properties. This is joint work with Toby Kenney.

Tuesday, December 1, 2009
Toby Kenney, Categories as monoids in Span (Joint with R. Paré)
Abstract: In the previous talk, R. Paré explained the correspondance between categories and quantales. Through the equivalence between Rel and the category of complete atomic boolean algebras and sup-homomorphisms, this correspondance can also be looked at as a correspondance between categories and monoids in Rel. Using this perspective, we will identify which quantales come from categories.

Tuesday, January 12, 2010
Bob Paré, Double Lawvere Theories
Abstract: I will first review Lawvere's theory of universal algebra and then show how to extend this to first order theories.

Tuesday, January 19, 2010
Richard Wood, Raney's Anonymous Relation (joint work with Toby Kenney)
Abstract: No, Virginia, 'The Man with no Name' was not part of Raney's extended family. Raney wrote a\rho b, for a and b in a complete lattice L, in case (\forall downsets S of L)(b\le \/S ===> a\in S). Raney gave several characterizations of completely distributive lattices, at least two of which involve \rho (which he did not otherwise name and which has often been called Raney's anonymous relation). For example, L is completely distributive if and only if (\forall b\in L)(b\le \/{a|a\rho b}). Much later, Rosebrugh and Wood wrote a<<b for a\rho b and read it as 'a is totally below b'.

The totally below relation can be defined for any (pre)ordered set (X,\le) and statements such as the (b\le \/{a|a<<b}) above continue to make sense without assumimg that any suprema exist. Thus one can speak of a completely distributive ordered set without knowing of any infima distributing over suprema. It seems that complete distributivity really addresses a predisposition of an ambient ordered set that in the presence of infima and suprema is captured by a family of algebraic equations.

Although we have not written about it, it seems clear that some of our work should extend from {\bf 2}-cat to general V-cat, and in particular to cat=set-cat.

This work unfolded in the course of our study of tensor products of sup-lattices and applications of << to the description of tensor products will be given in a later talk.

Tuesday, January 26, 2010
Toby Kenney, Idempotent relations, and their connection with CCD lattices and tensor products (Joint work with Richard Wood)
Abstract: Last week, Richard talked about how we can express concepts traditionally related to sup-lattices for arbitrary ordered sets. In particular, he discussed the totally below relation for these sets. The totally below relation is not in general reflexive. However, for certain kinds of ordered sets, it has the interpolation property that if x<<z then there is a y such that x<<y<<z. In this talk, we will look more closely at relations that are transitive and interpolative. These relations are closely connected to CCD lattices, and in this talk we explain more aspects of this connection. Using this, we give a construction of the tensor product of CCD lattices.

Tuesday, February 9, 2010
Bob Paré, Finite Products in Double Categories
Abstract: There are a number of possible definitions of finite products in a double category and only practice can sort out their relative merit. After examining the situation for ordinary categories we move on to the double case. Various definitions will be proposed and examples examined.

Tuesday, February 16, 2010
Bob Paré, Families in Double Categories
Abstract: We introduce a double category version of the category of families in a category. We then study the various properties of this double category. In particular it suggests a rather general notion of coproduct in this context.

Tuesday, March 2, 2010
Susan Niefield, A Generalized Glueing Construction

Tuesday, March 30, 2010
Bob Rosebrugh, Partial lenses and universal updates
Abstract: Database view updating can be seen as a lifting problem, so it is not surprising that fibrations play a role. In this talk we extend work of the authors linking these concepts.

Let C be a category with products. For an object B, the sum functor C/B-->C is a left adjoint, and an algebra (G:E-->B,P) for the generated monad on C/B is exactly the same thing as Pierce's notion `lens'. They are essentially projections.

When C = Cat a lens G:E-->B is an opfibration. On the other hand, taking the projection (G,1_B) --> B from the comma category is the functor part of a monad on Cat/B. An algebra for (-,1_B) provides a good notion of a "partial lens". An opfibration is such. Furthermore, an opfibration has "universal translations". These provide a universal solution to the view updating problem when G = W*:Mod(E) --> Mod(V) for a view (sketch morphism) W:V-->E in the Sketch Data Model.

This is joint work with Michael Johnson and Richard Wood

Tuesday, April 6, 2010
Octavio Malherbe, Partially traced categories and paracategories
Abstract: It is not always true that an arbitrary endomorphism can be traced in a symmetric monoidal category. To deal with this kind of situation Abramsky, Blute, and Panangaden, Blute, Cockett, and Seely and also Haghverdi and Scott extended the study of abstract properties of the trace in category theory that was begun in the work of Joyal, Street and Verity. This extension, through the notion of partially traced category, especially in the case of Haghverdi and Scott, was more focused on clarifying the computational aspects arising from Girard's Geometry of Interaction. In this talk, we consider Freyd's notion of a paracategory. We prove that every partially traced category (in the sense of Haghverdi-Scott) can be faithfully embedded in a symmetric monoidal compact category via an extended version of the construction made by Joyal, Street and Verity and the notion of a strict symmetric compact paracategory.