Richard Wood, Frobenius objects in cartesian bicategories
Abstract: For any object X in a cartesian bicategory #B, there is a canonical 2-cell
X \ox d
X \ox X -------------> X \ox X \ox X
| |
| |
d* | \delta | d* \ox X
| ==========> |
| |
v v
X -------------------> X \ox X
d
namely the mate of the isomorphism witnessing coassociativity of the
comonoid structure d:X ---> X \ox X, d being the diagonal map (left
adjoint). X is said to be {\em Frobenius} if \delta is invertible.
There is currently considerable interest in Frobenius objects in mere monoidal categories (see the book by Joachim Kock) and in the context of quantum computing (CT07 talk by Bob Coecke). It seems likely that in many cases such Frobenius structures are obtained by forgetting the 2-cells of a cartesian bicategory (and making identifications).
Frobenius objects were studied in cartesian locally ordered bicategories by Carboni and Walters [1987]. Without the general definition of cartesian bicategory at hand, we knew then that Frobenius objects in the bicategory #prof of categories and profunctors are groupoids but the proof has not been published. In this talk that proof will be given together with such progress as we have made towards proving that:
If X is a Frobenius object in a cartesian bicategory #B then, for any object T, Map(#B)(T,X) is a groupoid. (Joint work with Bob Walters.)
Monday, August 27, 2007
Travis Squires, Profunctors and Distributive categories
Abstract:
We attempt to formulate the correct definition of a profunctor between
distributive categories. We call such a profunctor a distributor.
Some properties of profunctors are reviewed and the analogous properties
are examined for distributors. In particular there is a relationship
between distributive functors and distributors, similar to a relationship
holding between functors and profunctors. Given a distributor we construct
some distributive categories and distributive functors. Next we build a
distributive category whose objects are polynomials and provide an evaluation
functor along with an appropriate universal property. In an attempt to model
the situation in algebraic geometry we construct adjoint functors and hope to
generalize the coordinate ring construction using distributors. To conclude,
some ideas of how one might proceed along these lines are discussed.
Monday, August 27, 2007
Alex MacLeod, A Computation of the E2 Term of the
Complex K-theory Adams Spectral Sequence
Abstract:
The Adams spectral sequence is an algebraic device used to compute stable
homotopy groups of spaces by means of a given cohomology theory, in our case
complex K-theory. We introduce the necessary algebraic background to study
K*K the Hopf algebroid of co-operations on complex K-theory, before explicitly
computing the E2 term of the complex K-theory Adams spectral sequence.
Tuesday, September 11, 2007
Toby Kenney, Difunctional Relations, Diadjunctions, and Diads I
Abstract:
A relation R is called d i f u n c t i o n a l if it satisfies RR^oR=R,
where R^o is the opposite relation. Difunctional relations were introduced
in 1948 by J. Riguet.
Recall that We can view fucntional relations as those relations that have right adjoints; in the same way, we can ask, what happens if we replace R^o by a general S, and work in an arbitrary bicategory, rather than just Rel. We call the ensuing concept a diadjunction. Diadjunctions have the advantage that unlike adjunctions, they do not have a handedness - There are not left diadjunctions and right diadjunctions. On the other hand, they don't have many of the nice properties of adjunctions, e.g. uniqueness of adjoints, preservation of limits, etc.
The interesting generalisation seems to be when we generalise monads to what we call diads. It turns out that most of the constructions we perform on monads, e.g. Kleisli category, Eilenberg-Moore category are actually constructions on the diads we get from them.
Tuesday, September 18, 2007
Dorette Pronk, Model Structures on DoubleCat
Tuesday, September 18, 2007
Margaret Beattie, Hopf algebras with a coalgebra projection
Abstract:
Let $A$ be a Hopf algebra over a field $k$ and suppose there is a
Hopf algebra map $ \pi$ from $A$ to a sub Hopf algebra $H$
that splits the inclusion. Then $A$ is a Radford biproduct $R
\times H$ where $R$ is the algebra of $\pi-$ coinvariants. Although $R$
is not a Hopf subalgebra of $A$, $R$ is a Hopf algebra in a different
category: the category of Yetter-Drinfeld modules over $H$.
The theory of Radford biproducts is key to the work of Andruskiewitsch and Schneider in their project of classification of finite dimensional pointed Hopf algebras. In this talk, I will attempt a brief overview of their work and hopefully at the end describe some recent work of Ardizzoni, Menini et al on what can be said if $\pi$ is not a Hopf algebra projection,
Tuesday, October 2, 2007
Bob Rosebrugh, Database views, lenses and monads
Abstract:
Views are important in database system design and management. The view
update problem: `when can a change in a view state be extended to the
total database state?' has been much studied. An early approach treated
states as unstructured sets and advocated views with `constant
complement', although `constant factor' would have been a better name.
Recently the `lens' concept and a simple monad have elucidated the
situation. An order-based approach with an ordered set of states provides
a generalization though still requiring complements. When states are a
category of models, a less narrow view of updatability emerges. (joint
with Michael Johnson and Richard Wood)
Tuesday, October 9, 2007
Toby Kenney, Difunctional Relations, Diadjunctions and Diads (cont.)
Abstract:
Last time we defined diadjunctions and diads, and dialgebras for a diad.
We showed that given a monad (T,\eta,\mu), dialgebras for (T,\eta_T,\mu)
are exactly the same thing as algebras for the monad (T,\eta,\mu). Now we
will look at some special types of diads, and the corresponding
dialgebras. We will also look at the Kleisli category for a diad.
We will illustrate some of this theory for the diad (T,T\eta,\mu) for a monad (T,\eta,\mu), where we get some well-known dialgebras.
Tuesday, October 16, 2007
Toby Kenney, Difunctional Relations, Diadjunctions & Diads III
Finite-limit Preserving Diads in Topoi
Abstract:
It is a well-known fact that the category of coalgebras for a finite-limit
preserving comonad on a topos is a topos. It is also well-known that the
category of algebras for an idempotent monad is a topos. In both these
cases, the category of coalgebras or algebras is the category of
dialgebras for the corresponding diad.
The diad we get from either a comonad or an idempotent monad is a left diad. We can extend the two well-known results above to hold for any distributive diad -- i.e. the category of dialgebras for a distributive finite-limit preserving diad on a topos is again a topos.
Tuesday, October 23, 2007
Dorette Pronk, Translation Groupoid Representations for Orbifolds
Abstract:
Orbifolds consist of an underlying paracompact Hausdorff space with an
atlas of charts, just like manifolds, except that the charts for an
orbifold consist of an open subset of Euclidean space with an action of a
finite group, such that the quotient by this action is an open subset of
the underlying space of the orbifold.
If one has a manifold with an action of a compact Lie group such that the isotropy groups are finite, this can be shown to be an orbifold by using the differentiable slice theorem. Orbifolds that can be written in this form are called representable. Satake, who first introduced orbifolds in 1956, showed that if all the groups act effectively, the orbifold is representable.
In my thesis work I showed how one can use groupoids to obtain a category of orbifolds with morphisms which behave well with respect to homotopy theory. Representable orbifolds are precisely those orbifolds which have a representation by a translation groupoid.
For representable orbifolds we would like to use techniques from equivariant homotopy theory to obtain orbifold homotopy invariants, which would characterize the orbifold up to homotopy. The problem is that representations for orbifolds as a quotient of a manifold by the action of a compact Lie group are only unique up to essential equivalence.
In this talk I will discuss how groupoid representations for orbifolds lead to a definition of Bredon cohomology for orbifolds. This is joint work with Laura Scull from UBC.
Tuesday, October 30, 2007
Dorette Pronk, Translation Groupoid Representations for Orbifolds (continued)
Abstract:
Orbifolds consist of an underlying paracompact Hausdorff space with an
atlas of charts, just like manifolds, except that the charts for an
orbifold consist of an open subset of Euclidean space with an action of a
finite group, such that the quotient by this action is an open subset of
the underlying space of the orbifold.
If one has a manifold with an action of a compact Lie group such that the isotropy groups are finite, this can be shown to be an orbifold by using the differentiable slice theorem. Orbifolds that can be written in this form are called representable. Satake, who first introduced orbifolds in 1956, showed that if all the groups act effectively, the orbifold is representable.
In my thesis work I showed how one can use groupoids to obtain a category of orbifolds with morphisms which behave well with respect to homotopy theory. Representable orbifolds are precisely those orbifolds which have a representation by a translation groupoid.
For representable orbifolds we would like to use techniques from equivariant homotopy theory to obtain orbifold homotopy invariants, which would characterize the orbifold up to homotopy. The problem is that representations for orbifolds as a quotient of a manifold by the action of a compact Lie group are only unique up to essential equivalence.
In this talk I will discuss how groupoid representations for orbifolds lead to a definition of Bredon cohomology for orbifolds. This is joint work with Laura Scull from UBC.
Tuesday, November 6, 2007
Bob Paré, An Indexed Category Revival I
Abstract:
In the 30 years since Dietmar and I wrote our paper on indexed categories,
much has been understood more clearly. Yet there remains much to be done.
We have here the almost perfect group to carry this out. I'm thinking of
indexing by manifolds, topological spaces, coalgebras, measure spaces, and
other space like structures. From a logical point of view, Martin-Lof type
theory and Vopenka set theory are particularly relevant. In another direction,
enriched categories also play a crucial role. These are only a few topics
which enter into the general theory of indexed categories.
In this first talk I'll give some motivation together with basic definitions.
Tuesday, November 13, 2007
Bob Paré, An Indexed Category Revival II
Abstract:
I will give some examples of indexed categories, define indexed functors
and natural transformations. I'll also discuss fibrations.
Tuesday, November 20, 2007
Bob Paré, An Indexed Category Revival III
Abstract:
I will discuss completeness and smallness.
Tuesday, November 27, 2007
Geoff Cruttwell, Change of Base for Enriched Categories
Abstract:
One of the basic facts of enriched category theory is that any weak
monoidal functor N: V -> W induces a corresponding "change of base"
2-functor N* between V-categories and W-categories. However, there are
still a number of questions about this 2-functor. For example, if X is a
V-monoidal V-category, is N*X a W-monoidal W-category? What if X is
V-compact closed? Perhaps even more basic, we know that N* takes
V-functors to W-functors, but does it take V-profunctors to W-profunctors?
While the answers to these questions are relatively straightforward, the process of investigating them reveals certain interesting aspects of higher-dimensional category theory.
Tuesday, December 5, 2007
Peter Selinger, Completeness for dagger compact closed categories
Abstract:
Hasegawa, Hofmann, and Plotkin recently showed that the
category of finite-dimensional real vector spaces is complete for
compact closed categories. This means that for any pair of different
arrows in the free such category, there always exists a compact closed
functor into finite-dimensional vector spaces that distinguishes
them. Or in logical terms: any equation that holds in finite
dimensional vector spaces holds in all compact closed categories.
I will give a slightly simpler proof of this result. I will also prove the completeness of finite-dimensional *complex* vector spaces for *dagger* compact closed categories.
Wednesday December 12, 2007
Richard Wood, Elementary disjoint universal sums
(R Rosebrugh and RJ Wood)
Abstract:
Let Q:E--->S be a fibration. To say that Q is also an opfibration is
to say that the diagonal E--->Q/S has a left adjoint, call it \Sigma.
But Q/S--->S is also a fibration over S and for \Sigma to meaningfully
provide sums for the S-indexed category Q:E--->S is for \Sigma to be
left adjoint to E--->Q/S in the 2-category of fibrations over S. This
additional requirement is equivalent to the Beck condition (BC). With
\Sigma:Q/S--->E providing sums it makes sense to ask if these are
disjoint (SD) and universal (SU). Our theorem, which was conjectured
by Bob Paré, says
(BC) & (SD) & (SU) <===> \Sigma:P/S--->E is lex
(Actually this last is quivalent to \Sigma:P/S--->E preserving pullbacks; preservation of the terminal object being automatic.)
While this result is far from new (we published it in 1986) it nevertheless fits well into the Indexed Category Revival and can probably be understood better now.
Tuesday, January 15, 2008
Toby Kenney, The Quantale of a Category
Tuesday, January 22, 2008
Benoit Valiron, More Quantum Stuff
Tuesday, January 29, 2008
Bob Paré, Seeing Double
Abstract:
Recently Walter Tholen and Gavin Seel have given talks
on how various topological categories can be usefully viewed as
the categories of lax algebras for certain monads. And every time,
I tell them they should be using double categories. So it's time
to see if there's anything in this.
I will survey the topic and see if I can make anyone else see double too.
Tuesday, February 5, 2008
Bob Paré, Double Triples
Abstract:
I will continue last week's talk.
Tuesday, February 12, 2008
Dorette Pronk, Double, double toil and trouble
Abstract:
Although we know that the category of small double categories is
closed under colimits, it is not always easy to describe these colimits.
In our work on model structures for the category of double categories,
we want to transfer model structures from the category of simplicial
objects in Cat. In order to do this we need to prove results about certain
pushouts in DoubleCat. In my talk I will briefly discuss the origin
of the "toil and trouble" (the Lemma from Kan on Transfer) and then
present a normal form for the cells in the pushout double category.
Tuesday, February 19, 2008
Dorette Pronk, Double, double toil and trouble II
Abstract:
Although we know that the category of small double categories is
closed under colimits, it is not always easy to describe these colimits.
In our work on model structures for the category of double categories,
we want to transfer model structures from the category of simplicial
objects in Cat. In order to do this we need to prove results about certain
pushouts in DoubleCat. In my talk I will briefly discuss the origin
of the "toil and trouble" (the Lemma from Kan on Transfer) and then
present a normal form for the cells in the pushout double category.
Tuesday, March 4, 2008
Dorette Pronk, Double, double toil and trouble III
Abstract:
The proof.
Tuesday, March 11, 2008
Bob Rosebrugh, Compositional colimits and Kleene's theorem
Abstract:
Categories of spans and cospans of graphs are useful in the study of
algebras of processes. We are interested in describing systems constructed
from parts in two ways. The first is a geometrical description of the
configuration of the parts; the second is an algebraic description in
which the total system is an expression in the parts, in an algebra of
systems. The geometric viewpoint consists of a description of the system
as a (co)limit of subsystems, the (co)limit parametrized by some form of
graph. The algebras which allow a system to be described compositionally
are the well-supported compact closed (wscc) categories of spans and
cospans in a category whose objects are state spaces of systems. Kleene's
theorem says that the languages recognized by finite state automata are
precisely those obtained by closure of singletons under concatenation,
union and iteration. We will show how the results above may be interpreted
as an abstract Kleene theorem. (joint with R.F.C. Walters and N. Sabadini)
Tuesday, March 18, 2008
Richard Wood, Duals Invert
Abstract:
Recently, Day and Pastro introduced the notion of Frobenius
monoidal functor --- by which is meant a monoidal functor
(F,\phi,\phi_0):V--->W together with comonoidal structure
(\psi,\psi_0) satisfying
\psi.\phi=(1\ox\phi)(\psi\ox1):F(A\ox B)\ox FC--->FA\ox F(B\ox C)
\psi.\phi=(\phi\ox1)(1\ox\psi):FA\ox F(B\ox C)--->F(A\ox B)\ox FC
Amongst other things, they showed that Frobenius monoidal functors take
dual situations to dual situations and that if \tau:F==>G:V--->W, where
F and G are Frobenius and \tau is both monoidal and comonoidal, then, for
A in V with either a left dual or a right dual, \tau A:FA--->GA is
invertible.
Strong monoidal functors are Frobenius and it was always clear that strong monoidal functors send duals to duals. Ross Street had already observed that `duals invert' for \tau monoidal and comonoidal and F and G strong. He had also conjectured that the theorem of Walters and Wood `For A a Frobenius object in a cartesian bicategory \B and any X, \map\B(X,A) is a groupoid' can be seen as an instance of his monoidal result if the latter is internalized to a monoidal bicategory. The theorems of Day and Pastro provided impetus to carry out this program with strong (internal) arrows generalized to Frobenius arrows.
Joint work with Ross Street.
Thursday, March 20, 2008
Gabor Lukacs, Topological groups and rings of continuous functions
Tuesday, March 25, 2008
Richard Wood, Duals Invert (continued)
Abstract:
Recently, Day and Pastro introduced the notion of Frobenius
monoidal functor --- by which is meant a monoidal functor
(F,\phi,\phi_0):V--->W together with comonoidal structure
(\psi,\psi_0) satisfying
\psi.\phi=(1\ox\phi)(\psi\ox1):F(A\ox B)\ox FC--->FA\ox F(B\ox C)
\psi.\phi=(\phi\ox1)(1\ox\psi):FA\ox F(B\ox C)--->F(A\ox B)\ox FC
Amongst other things, they showed that Frobenius monoidal functors take
dual situations to dual situations and that if \tau:F==>G:V--->W, where
F and G are Frobenius and \tau is both monoidal and comonoidal, then, for
A in V with either a left dual or a right dual, \tau A:FA--->GA is
invertible.
Strong monoidal functors are Frobenius and it was always clear that strong monoidal functors send duals to duals. Ross Street had already observed that `duals invert' for \tau monoidal and comonoidal and F and G strong. He had also conjectured that the theorem of Walters and Wood `For A a Frobenius object in a cartesian bicategory \B and any X, \map\B(X,A) is a groupoid' can be seen as an instance of his monoidal result if the latter is internalized to a monoidal bicategory. The theorems of Day and Pastro provided impetus to carry out this program with strong (internal) arrows generalized to Frobenius arrows.
Joint work with Ross Street.
Tuesday, April 1, 2008
Richard Wood, Duals Invert (continued)
Abstract:
Recently, Day and Pastro introduced the notion of Frobenius
monoidal functor --- by which is meant a monoidal functor
(F,\phi,\phi_0):V--->W together with comonoidal structure
(\psi,\psi_0) satisfying
\psi.\phi=(1\ox\phi)(\psi\ox1):F(A\ox B)\ox FC--->FA\ox F(B\ox C)
\psi.\phi=(\phi\ox1)(1\ox\psi):FA\ox F(B\ox C)--->F(A\ox B)\ox FC
Amongst other things, they showed that Frobenius monoidal functors take
dual situations to dual situations and that if \tau:F==>G:V--->W, where
F and G are Frobenius and \tau is both monoidal and comonoidal, then, for
A in V with either a left dual or a right dual, \tau A:FA--->GA is
invertible.
Strong monoidal functors are Frobenius and it was always clear that strong monoidal functors send duals to duals. Ross Street had already observed that `duals invert' for \tau monoidal and comonoidal and F and G strong. He had also conjectured that the theorem of Walters and Wood `For A a Frobenius object in a cartesian bicategory \B and any X, \map\B(X,A) is a groupoid' can be seen as an instance of his monoidal result if the latter is internalized to a monoidal bicategory. The theorems of Day and Pastro provided impetus to carry out this program with strong (internal) arrows generalized to Frobenius arrows.
Joint work with Ross street.
Tuesday, June 17, 2008 (at MtA)
Alessandro Ardizzoni, U. Ferrara, Universal Enveloping Algebras of Braided Vector Spaces
Abstract:
Various attempts to find a proper generalization of the
notion of Lie algebra associated to a vector space $V$ endowed with a
non symmetric braiding $c$ appeared in the literature. In this
direction, we introduce and investigate a notion of braided Lie
algebra (and the associated universal enveloping algebra) which turns
out to be effective for the class of braided vector spaces $(V,c)$
whose Nichols algebra is obtained dividing out the tensor algebra
$T(V,c) $ by the two-sided ideal generated by its primitive elements
of degree at least two. One of the main applications of our
construction is the description, in terms of universal enveloping
algebras, of connected braided bialgebras whose associated graded
coalgebra is a quadratic algebra.
Tuesday, June 17, 2008 (at MtA)
Colin Ingalls, U.N.B., Noncommutative Coordinate Rings of Stacks
Abstract:
We explore the relation between noncommutative algebras
and stacks. Let B=>A be a Hopf algebroid of commutative algebras,
or equivalently an affine stack. If A is a finitely generated projective
B-module we can make a noncommutative algebra out of Hom(B,A)
which has the same module category as the Hopf algebroid. We
show that hereditary orders over curves correspond to Deligne-Mumford
stacky curves with trivial generic stabilizer via this construction.
Wednesday, June 18, 2008
Geoff Cruttwell, Change of Base and the Amazing Technicolour Double Categories
Abstract:
In this talk, I'll discuss how a certain change of base question can only
be resolved if one thinks of V-cat not as a 2-category, not as a
bicategory, but instead as a double category.