Tuesday, September 11, 2012
Toby Kenny, Graphical Composition
Abstract: Graphical composition was defined by H. Werner as a method for identifying which sublattices of partition lattices occur as congruences for some algebra structure on the underlying set. Despite the obvious importance of this, graphical composition has not received so much attention, largely because of the difficulty in computing it. We recast graphical composition from a categorical perspective, based on the construction of the quantale of category, and use this approach to generalise the notion to more general quantales. We hope this level of generality will allow us to obtain important insights about graphical composition.

Tuesday, September 18, 2012
Geoff Cruttwell, Mount Allison, The "fundamental theorem of tangent structures"
Abstract: Rosicky's notion of tangent structure is an axiomatization of the central properties of the tangent bundle functor of differential geometry. However, we will see that it also provides the missing link between the Cartesian differential categories of Blute, Cockett and Seely, and the synthetic differential geometry of Kock and Lawvere. In particular, we'll show how a single theorem implies that the axioms for both theories are contained in the axioms for tangent structure.

Tuesday, September 25, 2012
Margaret Beattie, Mount Allison, Classifying low dimensional Hopf algebras with the Chevalley property
Abstract: The general problem of classifying all Hopf algebras of a given finite dimension over the complex numbers can be tackled by looking at classes of Hopf algebras. For example, all Hopf algebras of p, 2p or p^2 dimension, p a prime, are classified. All semisimple Hopf algebras of dimension pq, p < q primes, are group algebras or duals of group algebras. The classification of pointed Hopf algebras with abelian group of points has been done through the program of Andruskiewitsch and Schneider and this research group is working on understanding the pointed Hopf algebras with nonabelian group of grouplikes.

A Hopf algebra is said to have the Chevalley property if its coradical is a sub-Hopf algebra. This is more general than saying that the Hopf algebra is pointed but one would think that this should be tractable given a knowledge of the semisimple Hopf algebras. In this talk we define and discuss the Chevalley property for a tensor category and then look at Hopf algebras of some small dimensions with the Chevalley property.

Tuesday, October 2, 2012
Bob Paré, Multi-valued functors
Abstract: Diers defines what it means for a functor to have a multi-adjoint but stops short of saying what sort of thing such a multi adjoint might be. Adamek and Rosicky as well as Makkai and Par?? use the multi-limits but don't say where they live either. Lack and Street, in their paper on formal monads, introduce the appropriate bicategory structure for Lawvere's partial functors but don't relate it to multi-adjoints. We put the two together and explore the consequences.

Tuesday, November 6, 2012
Dorette Pronk, Weakly Globular Double Categories

Tuesday, November 13, 2012
Dorette Pronk, Weakly Globular Double Categories - Continued

Tuesday, November 20, 2012
Dorette Pronk, Weakly Globular Double Categories - Continued

Tuesday, November 27, 2012
Dorette Pronk, Weakly Globular Double Categories - Continued - Continued

Tuesday, December 4, 2012
Dorette Pronk, Weakly Globular Double Categories - the final frontier

Tuesday, January 29, 2013
Robert Paré, Kaput Profunctors
Abstract: Over 40 years ago James Kaput [1] introduced a generalization of left adjoint to a functor. This is a bit more general than Diers' multi-adjoints I discussed in the first term. I will introduce his notion, compare it with multi adjoints, give examples and reformulate it in terms of profunctors.

1. Kaput, James, Locally Adjunctable Functors, Illinois J. Math. Vol. 16, Issue 1 (1972), 86-94.

Tuesday, February 5, 2013
Peter Selinger, Approximations in quantum computing
Abstract: Consider the symmetric monoidal category of finite dimensional Hilbert spaces and unitary maps. Let V be a 2-dimensional Hilbert space, and consider the full subcategory whose objects are V^n, where n=0,1,2,... This category is obviously not finitely generated (in the algebraic sense), as it has uncountably many morphisms. However, it is well-known that there exist dense symmetric monoidal subcategories that are finitely generated. A popular set of generators is the so-called Clifford+T set of quantum gates. In the first talk, I will define these notions and give a self-contained introduction to this area. In the second talk, I will give an intrinsic characterization of the Clifford+T symmetric monoidal category. This is joint work with Brett Giles. In the third talk, I will discuss the problem of approximating 2x2-unitary operations up to given epsilon. For the last 17 years, the standard solution to this problem was the so-called Solovay-Kitaev algorithm, which achieved gate sequences of length K*log^c(1/epsilon), where c > 3. I will present a new efficient number theoretic algorithm that achieves length K + 4*log(1/epsilon).

Tuesday, February 12, 2013
Peter Selinger, Approximations in quantum computing - Continued
Abstract: Consider the symmetric monoidal category of finite dimensional Hilbert spaces and unitary maps. Let V be a 2-dimensional Hilbert space, and consider the full subcategory whose objects are V^n, where n=0,1,2,... This category is obviously not finitely generated (in the algebraic sense), as it has uncountably many morphisms. However, it is well-known that there exist dense symmetric monoidal subcategories that are finitely generated. A popular set of generators is the so-called Clifford+T set of quantum gates. In the first talk, I will define these notions and give a self-contained introduction to this area. In the second talk, I will give an intrinsic characterization of the Clifford+T symmetric monoidal category. This is joint work with Brett Giles. In the third talk, I will discuss the problem of approximating 2x2-unitary operations up to given epsilon. For the last 17 years, the standard solution to this problem was the so-called Solovay-Kitaev algorithm, which achieved gate sequences of length K*log^c(1/epsilon), where c > 3. I will present a new efficient number theoretic algorithm that achieves length K + 4*log(1/epsilon).

Tuesday, February 19, 2013
Peter Selinger, Clifford+T approximation - the proof
Abstract: Consider the symmetric monoidal category of finite dimensional Hilbert spaces and unitary maps. Let V be a 2-dimensional Hilbert space, and consider the full subcategory whose objects are V^n, where n=0,1,2,... This category is obviously not finitely generated (in the algebraic sense), as it has uncountably many morphisms. However, it is well-known that there exist dense symmetric monoidal subcategories that are finitely generated. A popular set of generators is the so-called Clifford+T set of quantum gates. In the first talk, I will define these notions and give a self-contained introduction to this area. In the second talk, I will give an intrinsic characterization of the Clifford+T symmetric monoidal category. This is joint work with Brett Giles. In the third talk, I will discuss the problem of approximating 2x2-unitary operations up to given epsilon. For the last 17 years, the standard solution to this problem was the so-called Solovay-Kitaev algorithm, which achieved gate sequences of length K*log^c(1/epsilon), where c > 3. I will present a new efficient number theoretic algorithm that achieves length K + 4*log(1/epsilon).

Tuesday, March 5, 2013
Geoff Cruttwell, An introduction to quasicategories
Abstract: Like toposes, the quasicategories of Joyal can be viewed in different ways. One viewpoint is that they give a framework to understand "higher sheaves". Another views them as a half-way point between categories and infinity-categories. Another viewpoint is that they are the minimal conjunction of the notions of topological space and of category.

While I am no expert on quasicategories, while trying to understand them, I've found them surprisingly concrete, given their higher-categorical nature. Thus, in this talk I'll give a brief overview of the definition of quasicategories and some ideas of how one can work with them.

Tuesday, March 12, 2013
Alanod Sibih, Orbifold Atlas Groupoids - Thesis Presentation
Abstract: We study orbifolds and strong maps of orbifolds. We begin with introducing a representation for orbifolds that consists of internal categories in the category of topological spaces. These categories are built from atlas charts and chart embeddings without equivalence relation. They represent orbifolds and atlas maps, but do not work well for general strong maps. We generalize the notion of category of fractions to internal categories in the category of topological spaces. We find its universal property for an internal category in the category of topological spaces. We apply this to the atlas category to obtain an atlas groupoid. We give a description of strong maps of orbifolds and the equivalence relation on them in terms of atlas groupoids. We define paths in orbifolds as strong maps. We use our construction to give an explicit description of the equivalence classes on such paths in terms of charts and chart embeddings.

Tuesday, March 19, 2013
Rick MacLeod, Higher Algebra
Abstract: Higher dimensional categories have been investigated for decades. The definitions involve strict, weak (pseudo), and lax notions as well as biased versus unbiased presentations of these categories. However, aside from monads and operads there does not seem to be a coherent (pun intended) examination of the algebraic theories in these higher categories. With the aid of "Terminal Object Constructions" I present such an examination generalizing monads and showing operads and multi-categories as examples of "dyads". I will provide other examples and then state a way of obtaining algebraics in all dimensions. As a bonus I deduce the natural simplicial theory of ω-Categories.

Tuesday, March 26, 2013
Rick MacLeod, Higher Algebra - Continued
Abstract: Higher dimensional categories have been investigated for decades. The definitions involve strict, weak (pseudo), and lax notions as well as biased versus unbiased presentations of these categories. However, aside from monads and operads there does not seem to be a coherent (pun intended) examination of the algebraic theories in these higher categories. With the aid of "Terminal Object Constructions" I present such an examination generalizing monads and showing operads and multi-categories as examples of "dyads". I will provide other examples and then state a way of obtaining algebraics in all dimensions. As a bonus I deduce the natural simplicial theory of ω-Categories.

Tuesday, April 16, 2013
Peter Selinger, Clifford+T approximation - the proof
Abstract: Consider the symmetric monoidal category of finite dimensional Hilbert spaces and unitary maps. Let V be a 2-dimensional Hilbert space, and consider the full subcategory whose objects are V^n, where n=0,1,2,... This category is obviously not finitely generated (in the algebraic sense), as it has uncountably many morphisms. However, it is well-known that there exist dense symmetric monoidal subcategories that are finitely generated. A popular set of generators is the so-called Clifford+T set of quantum gates. In the first talk, I will define these notions and give a self-contained introduction to this area. In the second talk, I will give an intrinsic characterization of the Clifford+T symmetric monoidal category. This is joint work with Brett Giles. In the third talk, I will discuss the problem of approximating 2x2-unitary operations up to given epsilon. For the last 17 years, the standard solution to this problem was the so-called Solovay-Kitaev algorithm, which achieved gate sequences of length K*log^c(1/epsilon), where c > 3. I will present a new efficient number theoretic algorithm that achieves length K + 4*log(1/epsilon).