Robin Cockett, Free restriction categories
Abstract: Restriction categories are to partial maps as allegories are to relations: they provide a more convenient completely algebraic setting in which one can argue about partial maps. As restriction categories are algebraic there are "free" restriction categories and these correspond to free categories of partial maps.
It is possible to describe the free restriction category on a graph and on a category quite explicitly. Thus, it becomes interesting to ask whether it is possible to give similar explicit descriptions of other varieties of restriction categories. For example "classified" restriction categories correspond to partial map categories with a partial map classifier: what do the free categories with a partial map classifier look like? Range restriction categories correspond to the partial map categories of categories with an M-stable factorization ...
Tuesday, September 21, 2004
Robin Cockett, Free restriction categories (continued)
Tuesday, September 28, 2004
Robin Cockett, What are classifiers for general systems of maps?
Abstract:
The idea of partial map classification is well-known, but is it part of
a more general theory of classification for arbitrary maps? The purpose
of the talk is to suggest that there is such a general theory and I will
develop what seem to be its basic results. In particular, I shall
describe its link to partial map classification and to exponentiable
maps.
In addition, if there is time, I wish to take a quick look at some more exotic examples of this notion of general map classification arising from joint work with Richard Wood on the category of multi-categories.
Tuesday, October 5, 2004
Richard Wood, Indexing specializes to both variation and enrichment
Tuesday, October 12, 2004
Robin Cockett & Dorette Pronk, Orbifolds and all that ...
Abstract:
The idea of orbifolds seems to be a generalization of that of manifold: is
it the case, therefore, that there is a generalization of the manifold
construction which gives orbifolds? In trying to provide a categorical
answer to this question one must also answer the question of what the
maps beween orbifolds are ... which is not universally accepted to start
with!
The wonders of categorical machinery is that sometimes constructions suggest what the maps should be ... but do we have all the constructions right? Come and find out, even if you have no idea what a manifold or an orbifold is, we will start with the basic definitions.
Tuesday November 2, 2004
Bob Paré, The Pathology of Double Categories I
Abstract:
In this series of talks I will report on joint work with Robert Dawson and
Dorette Pronk where we study in detail two constructions for double
categories. Each produces a double category of paths with different
universal properties. Along the way we introduce various interesting
constructions. Our goal is to get a better understanding of our \Pi_2
construction and generalize it.
The first talk will be elementary and will serve as a warm-up for the rest.
Tuesday November 9, 2004
Bob Paré, The Pathology of Double Categories II
Abstract:
I will continue the study of paths in a category. Then I will introduce
double categories, double functors and lax morphism. At this point I
will be able to formulate the problem we are trying to solve.
Tuesday November 16, 2004
Bob Paré, The Pathology of Double Categories III
Abstract:
This week I will introduce lax morphisms of double categories. The
search for the universal lax morphism will lead to the double category
of paths in a double category. This gives a comonad (in fact a KZ one).
We will investigate the Kleisli and Eilenberg-Moore categories, time
permitting.
Tuesday November 23, 2004
Richard Wood, The Centre and the Core of a symmetric monoidal category
Abstract:
Peter Freyd has introduced the {\em core} of a category S with
finite products -*- as an object c in S, together with a natural
transformation u:(-)*c--->(-) which is universal for such data.
Universal means that any natural t:(-)*a--->(-) factors through
u via a unique arrow a--->c. Noting that naturality of t is
equivalent to extra-naturality of a--->(-)^(-), leads one to
conjecture that the core, given by \int_x x^x, is the (internal)
centre of S. It is not.
In spite of the triple x in \int_x x^x, this talk is suitable for all ages.
Tuesday November 30, 2004
Mitja Mastnak, About Drinfel'd-Yetter Modules
Abstract:
YD-module over a Hopf algebra H is a vector space V, together with an
H-module structure mu: HxV->V, mu(h,v)=hv and an H-comodule structure
delta: V->HxV, delta(v)=v_{-1}xv_{0} that satisfy the braiding condition
delta(hv)= h_1v_{-1}S^{-1}(h_3)xh_2 v_0. These structures play an
important role in various constructions of (pointed) Hopf algebras.
I will present some classification results for YD-modules over certain types of Hopf algebras. Many results are obtained by borrowing from the vast toolbox of the representation theory for finite groups. This is joint work with L. Grunenfelder.
Tuesday January 18, 2005
Gabor Lukacs, T-sequences in Topological Groups
Abstract:
Given an infinite set X, a sequence {x_n} and a point x_0 in X, one can
always find a Hausdorff topology on X in which x_n --> x_0. As in the case
of so many questions that are easy for topological spaces, once we switch
to the category of Hausdorff topological groups, suddenly it becomes a
very complicated one.
A sequence {a_n} in a discrete abelian group A is called a *T-sequence* if there is a Hausdorff group topology on A in which a_n --> 0. If {a_n} is a T-sequence on A, then there is a finest Hausdorff group topology on A with a_n --> 0, and we denote by A{a_n} the group A in that topology. It turns out that if {a_n} is not a T-sequence, there is a subgroup B of A such that {a_n + B} is a T-sequence in A/B, and B is maximal with respect to this property. Thus, we can extend the meaning of A{a_n} as A/B{a_n} whenever {a_n} is not a T-sequence in A.
The concept of T-sequence is only a particular case of the the more general notion of a T-filter. By introducing A{F} for a filter F on A, we obtain a method of defining "topological" relations on a group. The term "relations" is justified, because any relation in the group theoretic sense of the word can also be encoded this way. Therefore, this provides a topological extension of the classical meaning.
Tuesday February 8, 2005
Gabor Lukacs, The Bohr-compactification and T-sequences in topological groups
Abstract:
The category CompHaus of compact Hausdorff spaces (and their continuous
maps) is a reflective subcategory of Top. The spaces X whose reflection
X --> beta X is an embedding are precisely the Tychonoff spaces (= T_1 and
completely regular).
In case of topological groups, however, the situation is more complex,
because every T_0 group is Tychonoff. The category Grp(CompHaus) of
compact Hausdorff groups is reflective in Grp(Top), but the groups G whose
reflection rho: G --> bG is an embedding is a much narrower class that
those of the Tychonoff groups. Indeed, there are many T_0 groups that do
not admit any non-trivial continuous homomorphism into a compact Hausdorff
group.
A group G is precompact if for every neighborhood U of the identity there
is a finite subset F of G such that G=FU. In other words, if G can be
covered by finitely many translations of any neighborhood. It turns out
that Hausdorff precompact groups are precisely the groups such that
rho: G --> bG is an embedding.
There are, however, two more classes of groups that are of interest:
i) G is maximally almost periodic (MAP) if ker(rho)=1, i.e., if G admits
an injective continuous homomorphism into a compact Hausdorff group.
ii) G is minimally almost periodic (m.a.p.) if ker(rho)=G. In this case, G
admits no continuous homomorphisms into a compact Hausdorff group.
While many examples are known for both classes, in this talk we will use
the technique of T-sequences to present a group that is neither MAP nor
m.a.p.
Tuesday February 15, 2005
Bob Paré, Categorical Bagatelles
Abstract:
I will present a few of my favorite categorical trivialities, and if
you think that's a pleonasm, this is not the talk for you.
Tuesday February 22, 2005
Jeff Egger, On Frobenius algebras, quantales, monads and spaces
Abstract:
The notion of Frobenius algebra originally arose in ring theory, but
it is a fairly easy observation that this notion can be extended to arbitrary
monoidal categories. But, is this really the correct level of generalisation?
For example, when studying Frobenius algebras in the *-autonomous category Sup,
the standard concept using only the usual tensor product is less interesting
than a similar one in which both "tensor" and its de Morgan dual ("par") are
used. Thus we maintain that the notion of linear-distributive category (which
has both a tensor and a par, but is nevertheless more general than the notion
of monoidal category) provides the correct framework in which to interpret the
oncept of Frobenius algebra.
Tuesday March 8, 2005
Rob Milson, Introductory survey of synthetic differential geometry I:
Weil Algebras and Microlinearity
Abstract:
Synthetic differential geometry is an alternative formalization of
differential geometry based on the notion infinitesimal closeness.
The aim of this survey is to introduce the audience to some key
technical ideas underlying SDG and to provide at least one reasonable
example. A nice introduction to the SDG notion of infinitesimal is
the essay "Differential calculus and nilpotent real numbers", by
Anders Kock, available online at http://home.imf.au.dk/kock/real.PDF
The basic technique of SDG is to formalize the truncation Taylor series to polynomials by means of nilpotent elements. To allow truncations of various orders and "shape" we require appropriate nilpotent algebraic systems: these are the Weil algebras. The assumption of differentiability can then be encapsulated by an interesting and elegant axiom schema called microlinearity.
Tuesday March 15, 2005
Rob Milson, Introductory survey of synthetic differential geometry II:
The Lie bracket
Abstract:
The first non-trivial operation in the theory of differentiable
manifolds is that of the Lie bracket of two vector fields. This
natural operation is a first-order differential operator that combines
two vector fields to make another vector field. The resulting vector
fields measures the non-commutativity of the operation's arguments.
Classically, vector fields on manifolds were referred to as
"infinitesimal transformations" and formed the core of many classical
arguments in differential geometry. An analytical formalization of
differential geometry cannot accommodate such synthetic arguments. In
SDG, by contrast, they are put on a solid theoretical basis. It
should therefore prove instructive to see how SDG defines the Lie
bracket as a composition of infinitesimal transformations.
Tuesday March 22, 2005
Bob Paré, C-infinity Rings
Abstract:
C-infinity rings are algebras in which all C-infinity functions
can be interpreted. The category of such is an algebraic category and
so enjoys many good properties. We will see that many of the ideas of
synthetic geometry can be interpreted in it. But there are some desirable
features that it doesn't have, e.g. it's not a topos. We then pass to
presheaves which is better. We will hint at a category of sheaves which
is even better, allowing the results of SDG to be proved by "set theoretical"
means and interpreted in classical differential geometric terms.
Tuesday March 22, 2005
Bob Paré, C-infinity Rings - Continued
Tuesday May 10, 2005
Richard Wood, Cartesian Bicategories II
(Joint work with A. Carboni and G.M. Kelly)
Abstract:
In it's purest form in category theory, the adjective cartesian
refers to finite products. Of course there are many extensions of that
terminology to include various scenarios involving finite limits other
than finite products. It won't surprise readers then to learn that a
cartesian bicategory may fail to have finite products. In fact, Rel E
for E regular, Idl E for E exact, Span E for E with finite limits, and
Prof E for E a topos are all examples of cartesian bicategories which, for most values of E, fail to have finite products.
However, all these examples are bicategories, let us write one as B, for which the hom categories B(X,A) have finite products in the usual sense and the subbicategory of left adjoints, often called Map B, has finite products in the sense appropriate to bicategories (meaning that composition with projections mediates an equivalence of categories --- rather than an isomorphism of sets). A bicategory with these properties is, perhaps annoyingly, said to be precartesian.
Any precartesian B admits canonical lax functors \ox and I and canonical lax transformations t and u:i 1--->I as below.
\ox
B x B ---------------> B
^ ^ ^ ^ I
| | |^ \
i x i | t| i| \u 1
| | | \ /
| | v 1
Map B x Map B ---------> Map B
x
Here x:Map B x Map B--->Map B<---1:1 are the pseudofunctors that provide `finite products' for Map B and i is the inclusion. (We write \ox for the symbol that LaTeX calls \otimes. Pronounce it as `tensor'.)
The lax functors \ox and I, like any lax functors, include `constraints' \tilde\ox: (T\ox U)(R\ox S) ---> (TR)\ox (US) \ox^\circ: 1_{X\ox Y} ---> 1_X \ox 1_Y \tilde I: I1_*.I1_* ---> I1_* I^\circ: 1_{I*} ---> I1_*
The precartesian B is said to be cartesian if all four of these (with the first three sufficing) enjoy the property of invertibility. In this event the lax natural t and u become invertible pseudonatural transformations.