Richard Wood, The wavy arrows of a totally cocomplete category (Joint work with Francisco Marmolejo and Bob Rosebrugh)
Abstract: Every (co)complete lattice L admits a ``totally below relation'', << , called a ``way below relation'' by some, and one of the simplest characterizations of ``L is completely distributive'' is given by ``every element x of L is the sup of the set of all elements totally below x''.
Recently, we defined ``totally distributive'' categories to be those totally cocomplete categories B for which X, given by X-|Y:B--->Cat(B^op,set) has a left adjoint. It transpires that a merely totally cocomplete category already admits a hom-like functor W:B--->Cat(B^op,set) and, adapting Johnson and Joyal, we write \tilde B(b,a) for W(a)(b) and refer to the elements of this set as the ``wavy arrows from b to a''.
Many of the theorems of (constructively) completely distributive lattices now suggest counterparts for totally distributive categories. In addition to finding suitable formulations, the new challenges involve both coherence issues and size issues.
Tuesday, September 20, 2011
Peter Lumsdaine, Free monads via inductive types
Abstract:
An indispensable result of classical category theory is the fact that
any endofunctor on a complete category preserving suitably filtered
colimits admits an algebraically free monad. This provides many
desirable free objects; but the standard proof uses transfinite
iteration along ordinals, and hence does not immediately transfer to
constructive settings.
We recast the proof slightly ? iterating not along ordinals, but over certain freely-generated filtered categories, which may be constructed as inductive-inductive [sic] types (or, less directly, using ordinary inductive definitions). This proves the theorem constructively, and at the same time elucidates some of the content of the classical proof.
Tuesday, September 27, 2011
Micah MacCurdy, Tannaka Reconstruction of Hopf and Weak Hopf Algebras
Abstract:
The Tannaka construction associates to suitable monoidal functors
F: A ---> V algebraic objects in V; among other things, algebras,
coalgebras, bialgebras, Hopf algebras, and many structured versions of
such things, depending on the properties of F. Using a graphical
notation, we can give the (now easy!) proofs that these constructions
work--in the literature they are generally omitted. Also, we show that
the same constructions applied to Frobenius monoidal functors gives
weak Hopf algebras; and we see that the geometric properties of the
graphical notation for such functors correspond to the geometric
properties of the string diagrams which define weak Hopf algebras.
Tuesday, October 4, 2011
Bob Rosebrugh, Universal database view updates and a distributive law
Abstract:
We will briefly review database semantic modeling via sketches and how
this allows a statement of the "view update problem" and specification of
a universal solution.
We have shown that when a view functor on database states is a fibration, the view delete updates may be universally lifted to database state updates, and this generalizes the classical "constant complement" updating strategy and its formulation via the lens notion.
We now can show that simultaneous, compatible delete and insert updatability is guaranteed when the view functor is an algebra for the composite monad from a distributive law. The distributive law links the monads for fibrations and opfibrations. Moreover, it is sufficient that the view functor is both a fibration and opfibration and satisfies a Beck-Chevalley condition.
(joint with Michael Johnson)
Tuesday, October 11, 2011
Robert Dawson, Finite geometry: an opportunity for category theory?
Tuesday, October 18, 2011
Rick MacLeod, Categorical Fourier Transforms
Abstract:
Category Theory has been described as 'General Abstract Nonsense'. There are few
(aside trom CS) 'real world' applications of Category Theory. In so far as the
Quantum Theory can be viewed as 'real world' we present such an application.
Fourier Transforms are fundamental to the study of some physical systems - in particular the Quantum Theory. We present a very general picture of these transforms using the language of Category Theory. Along the way we must have at our disposable the notions of Measure space and Hilbert space. In this regard the mysterious Dirac Delta function plays a central role. We show that Fourier Transforms are manifestations of (and this warms the cockles of my heart) the fundamental concept of category theory - the universal arrow!
Tuesday, November 1, 2011
Peter Selinger, Partially traced categories
Abstract:
I will review Haghverdi and Scott's notion of "partially traced
monoidal category" and give several examples. I will show that every
partially traced category can be faithfully embedded in a totally
traced one, which yields a completeness theorem. This is joint work
with Octavio Malherbe and Phil Scott, and is based on Octavio
Malherbe's Ph.D. thesis.
Tuesday, November 15, 2011
Mitja Mastnak, Bialgebras and coverings
Abstract:
I will discuss the (bi)category of bialgebras, with
(partial) coverings and the idea of classifying bialgebras up to
covering equivalence. This is joint work with A. Lauve and fits into the
general scheme of Grunenfelder and Paré of using coalgebras instead of
sets as parameterizing objects.
Tuesday, November 22, 2011
Micah McCurdy, Weak Bialgebras and Weak Hopf Algebras
Abstract:
We give an introduction to weak bialgebras and weak Hopf algebras,
discussing the most basic examples: category algebras and groupoid algebras.
Tuesday, January 10, 2012
Roman Fric (Mathematical Institute of the Slovak Academy of Sciences), Probability theory - a categorical approach
Tuesday, January 24, 2012
Kamil Bradler, Crossing Tsirelson's bound with supersymmetric entangled states
Abstract:
We construct a class of supersymmetric entangled states which is used as
a nonlocal resource in the CHSH (Clauser-Horne-Shimony-Holt) game. The
maximal winning probability of the game corresponds to an expected value
known as Tsirelson's bound and no ordinary quantum-mechanical states can
perform better. In this talk I will show how a supersymmetric quantum
state used as a resource in the game can beat Tsirelson's bound.
The role of category theory for various superstructures will be briefly
discussed.
Tuesday, January 31, 2012
Peter Schotch, Reflections on the Algebra of Logic
Abstract:
Unlike other areas of mathematics, the algebra of logic seems to have
made very few advances since its inception in the work of Boole. In this
talk I attempt to found the claim that the _real_ algebra_ of logic is
categorical. I also suggest, ever so delicately, that other work which
attempts to characterize a logic as a category of a certain sort--as e.g
in the work of Lambek (Categories and Deductive Systems), and Lambek and
Scott, are on the wrong track in that they give undue priority to a kind
of logic. The same might be said about Topos theory which, in effect,
builds in the view that the algebra of logic is simply bounded
distributive lattice theory.
Part of my project is to redo the the algebra of logic as properly
categorical. This requires a view of the operators as limits which can
be maintained up to but not including modal logic. For the latter case I
am tinkering with the idea that necessity arises as a natural
transformation of truth.
Tuesday, February 7, 2012
Robert Paré, Polynomial Functors à la Macdonald
Abstract:
I will give an expository talk on polynomial functors defined on the category
of finite dimensional vector spaces. This is 35 years old but has intriguing
features from a more modern point of view.
Ref. I.G.Macdonald, Symmetric Functions and Hall Polynomials
Tuesday, February 14, 2012
Jeff Egger, How many categories of locally compact abelian groups are there?
Abstract:
In this talk, we shall consider the question "do Fourier transforms form a
natural transformation?". As it turns out, the answer is "yes, modulo an
embarrassing fact---namely, that I don't really have a handle on the source
category". If time permits, we shall also consider the Plancherel Theorem,
and the categorical can of worms that that opens.
Tuesday, February 21, 2012
Misha Kotchetov, MUN, Gradings on classical Lie algebras
Abstract:
Gradings by groups play an important role in the theory of Lie algebras and
their representations: for example, the Cartan decomposition of a semisimple
Lie algebra of rank r is a grading by the free abelian group of rank r. We are
interested in classifying gradings by arbitrary groups on simple Lie algebras.
A grading is called fine if it cannot be refined. The Cartan decomposition
mentioned above is a fine grading. Over an algebraically closed field, a
classification of fine gradings is known for matrix algebras, the algebra of
octonions, the exceptional simple Jordan algebra (characteristic different from
2) and the simple Lie algebras of the series A, B, C, D (characteristic zero
for D4 and different from 2 for all others) and of the exceptional types E6
(characteristic zero), F4 (characteristic different from 2) and G2
(characteristic different from 2, 3). I will present the classification of fine
gradings on the simple Lie algebras of the series A, B, C, D, and a recent
joint work with Alberto Elduque where we compute the Weyl groups of these
gradings (except for type D4).
Tuesday, March 6, 2012
Andrea Cesaro, Milan, Introduction to model categories
Abstract:
A brief introduction to the model categories and the building of homotopic
associated theories
Tuesday, March 13, 2012
Peter LeFanu Lumsdaine, The Hopf fibration, via logic
Abstract:
Topology is logic (that's the intentional "is")
Tuesday, March 20, 2012
Julien Ross, Interaction nets
Abstract:
Interaction nets are a model of computation based on Linear Logic's proof nets. I will
present the syntax of interaction nets and their denotational semantics.
Tuesday, March 27, 2013
Gábor Lukács, Automatic continuity and open mapping theorems of topological algebras
Abstract:
There are a number of interesting results in the literature stating
that certain algebraic and/or set-theoretic conditions on a map imply the
continuity of a map or its inverse. A few prominent examples are:
1. every group homomorphism of SO(3,R) into a compact group is continuous;
2. every bounded linear operation from a Banach space onto a Banach space
is open;
3. every *-homomorphism of C^*-algebras is continuous;
4. every continuous homomorphism from the group Z equipped with the p-adic
topology is open onto its image.
In this talk, we survey results of this nature, and present a solution to
the problem posed by Jeff Egger a few weeks ago concerning continuous
homomorphisms of locally compact abelian groups that preserve null sets
(a.k.a. Wendt maps).
Tuesday, April 3, 2012
Dorette Pronk, Bredon Cohomology With Local Coefficients
Abstract:
In this talk I will begin by discussing the classical concepts of cohomology
with constant coefficients and cohomology with local coefficients, and explain what
they show us about the topological space we are considering.
Then we will look at spaces with a continuous action of a fixed topological group G.
Cohomology has been generalized to this setting in two ways: via the Borel construction
and by Bredon's construction. I will briefly sketch the differences between the two and
then focus on Bredon's definition. His original definition was given for actions by a
discrete group and only considered constant coefficients. I will discuss how this has
been generalized to a cohomology with local coefficients for spaces with an action
by an arbitrary topological group. The new parts of this work are joint work with Laura Scull.
Tuesday, April 24, 2012
Dorette Pronk, Bredon Cohomology With Local Coefficients - Continued
Tuesday, May 1, 2012
Miklos Bartha, Memorial, Quantum Turing automata
Abstract:
A denotational semantics of quantum Turing machines
is defined in the dagger compact closed category of finite
dimensional Hilbert spaces. Using the Moore-Penrose generalized
inverse, a new additive trace is introduced on the restriction
of this category to isometries, which trace is carried over to
directed quantum Turing machines as monoidal automata. Using the
Joyal-Street-Verity Int construction, the resulting traced
monoidal category is embedded into a totally symmetric self-dual
compact closed category, which is then transformed into the indexed
monoidal algebra of undirected quantum Turing automata.