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).