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.