Tuesday, July 21, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 0*

**Abstract:** Since returning from Portugal, I've been developing a constructive
theory of effort spaces (which is what I call "generalised metric spaces in the
sense of Lawvere") and (quasi)uniform spaces. In this talk, I will review the
corresponding classical theory, and perhaps also sketch out how the subject
pertains to the rest of my current research project.

Tuesday, July 28, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 0.5*

**Abstract:** The more I think about last week's talk, the more dissatisfied
I become with how incomplete it was---so I propose to give a second
preliminary talk. This one will focus on examples, though I will also
give a rigorous definition of quasiuniform space, and discuss separation
axioms (in particular, regularity) in a bit more detail.

Tuesday, August 4, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 1.0*

**Abstract:** In the literature, one will find three equivalent defininitions
of "uniform space", all of which generalise to notions of "uniform locale"
that are also (at least classically) equivalent. I will pose a fourth
equivalent definition of "uniform space", whose natural generalisation to
locales is (even classically) inequivalent to the usual notion. It's
unclear whether the extra examples are of any interest, but I find this
definition of uniform locale easier to manipulate.

Tuesday, August 11, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 1.5*

**Abstract:** I will review the definitions of effort locale and
quasiuniform locale I gave last time; give some examples of
how they differ from more conventional definitions, and prove
some elementary results to (try to) illustrate why I think my
definitions are easier to work with.

Tuesday, August 18, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 2.0*

**Abstract:** I will continue exploring the relationship between fattening
operations, entourages, uniform covers, and efforts. (Sadly, symmetry
will have to wait 'til part 3.)

Tuesday, August 25, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 2.5*

**Abstract:** I will continue ...

Tuesday, September 1, 2015

Jeff Egger, *Effort locales and (quasi)uniform locales, part 3.0*

**Abstract:** I'll wrap up this series of lectures by trying to say
something sensible about the symmetry axiom. When I began in
July, this seemed like a minor obstacle, but it's proven to be
quite stubborn.

Tuesday, September 29, 2015

Gabor Lukacs, * Weakly complete spaces and the character groups of Hopf algebras
(joint work with Rafael Dahmen)
*

**Abstract:** The characters of a Hopf algebra (i.e., algebra homomorphisms into the
ground field) form a topological group with respect to the dual of the
comultiplication and the topology of pointwise convergence.

These talks are motivated by the problem of characterizing the topological groups that occur as a character group of some Hopf algebra.

The first talk in the series will focus the dual to the category of vector
spaces: the weakly complete topological vector spaces and their continuous
linear maps. This category turns out to be the natural habitat for the
character groups of Hopf algebras.

Tuesday, October 6, 2015

Gabor Lukacs, * Weakly complete spaces and the character groups of Hopf algebras (continued)
(joint work with Rafael Dahmen)
*

Tuesday, October 13, 2015

Peter Selinger, * Interacting Hopf Algebras
*

**Abstract:**I will report on recent work by Bonchi, Sobocinski, and Zanasi, and in particular Zanasi's Ph.D. thesis. In a nutshell, they use spans and cospans, and a general construction of Steve Lack, to give a presentation, by generators and relations, of the symmetric monoidal category of finite-dimensional vector spaces and linear functions (or relations), over any field. The equations take the form of two different Hopf algebra structures that interact in an interesting way.

Tuesday, October 20, 2015

Bob Paré, *Superspans
*

**Abstract:**I will discuss a generalization of the span construction based on a category and a "larger" category (supercategory) with a choice of pullbacks. This is a simple idea but of course it's the morphisms that are the interesting thing. We get tractable examples of lax or colax double functors which are not strong nor even normal. In a future talk I will show how this produces examples of non-trivial intercategories.

Tuesday, October 27, 2015

Peter Selinger, * Interacting Hopf Algebras (continued)*

Tuesday, November 10, 2015

Richard Wood, * The Characterization of Totally Distributive Categories*

**Abstract:** In this talk I will prove the following:

Theorem: For a total category \K, the following are equivalent:

i) \K is totally distributive;

ii) \K ~ i-Mod(1,\K^~);

iii) (\exists a Freyd/Street-small taxon T)(\K ~ i-Mod(1,T)).

Tuesday, November 17, 2015

Bob Paré, *Intercategories of Superspans*

**Abstract:** After a quick review of intercategories, I will give
a method of constructing non trivial and (hopefully) interesting
examples.

Tuesday, January 5, 2016

Richard Wood,

*Introduction to cartesian bicategories*

Tuesday, January 12, 2016

Dorette Pronk,

*Segal Factorization and General Composition in a Double Category*

**Abstract:**Arbitrary tilings of cells in a double category may not be composable to a single cell through repeated horizontal and vertical composition. The most well-known tilings that are not composable this way are the so-called pinwheel tilings. In earlier work Dawson and Pare have shown that under certain factorization conditions for individual cells it is possible to remove the obstruction and the resulting composition of tilings satisfies general associativity so that there is a unique composite cell for each tiling.

Weakly globular double categories satisfy a different type of factorization property, the Segal condition. This is a condition that requires the existence of certain cells that allow us to factor whole tilings in a particular way. In this talk I will discuss how this condition gives rise to an operation on tilings which combined with horizontal and vertical composition allows us to reduce each rectangular tiling to a unique single tile.

Tuesday, January 19, 2016

Dorette Pronk,

*Segal Factorization and General Composition in a Double Category - continued*

Tuesday, January 26, 2016

Rory Lucyshyn-Wright,

*Enriched algebraic theories and monads for a system of arities*

**Abstract:**Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category V with respect to a specified system of arities j:J \hookrightarrow V. Lawvere's notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted V-enriched J-cotensor theory, or J-theory for short. For suitable choices of V and J, such J-theories include the enriched algebraic theories of Borceux and Day, the enriched Lawvere theories of Power, the equational theories of Linton's 1965 work, and the V-theories of Dubuc, which are recovered by taking J = V and correspond to arbitrary V-monads on V. We identify a modest condition on j that entails that the V-category of T-algebras exists and is monadic over V for every J-theory T, even when T is not small and V is neither complete nor cocomplete. We show that j satisfies this condition if and only if j presents V as a free cocompletion of J with respect to the weights for left Kan extensions along j, and so we call such systems of arities eleutheric. We show that J-theories for an eleutheric system may be equivalently described as (i) monads in a certain one-object bicategory of profunctors on J, and (ii) V-monads on V satisfying a certain condition. We prove a characterization theorem for the categories of algebras of J-theories, considered as V-categories A equipped with a specified V-functor A -> V.

Tuesday, February 2, 2016

Rory Lucyshyn-Wright,

*Enriched algebraic theories and monads for a system of arities - continued*

Tuesday, February 23, 2016

Bob Rosebrugh,

*Spans of edit lenses*

**Abstract:**"Asymmetric" lenses provide a strategy to lift updates in a model domain along a morphism of model domains. The model domains may be sets, orders or categories. A "symmetric" lens between two model domains has state synchronization data and resynchronization operations. In previous work we showed that (certain equivalence classes of) spans of asymmetric lenses represent symmetric lenses for both the set-based and category-based domains. The more recently introduced edit lenses are almost category-based, but they have updates from a fixed monoid of "edits". Edit lenses had been defined in the symmetric case only. We have found a suitable notion of asymmetric edit lens which allows a representation of symmetric edit lenses by spans of asymmetric ones. This is joint work with Michael Johnson.

Tuesday, March 1, 2016

Jeff Egger,

*Non-Hausdorff topologies and polar decomposition: a divertissement*

**Abstract:**We show that a proper understanding of polar decomposition leads naturally to the Sierpinski space, and other non-Hausdorff spaces.

Tuesday, March 8, 2016

Jeff Egger,

*An update on effort locales and quasi-uniform locales*

**Abstract:**I present a solution to the symmetry problem that so vexed me last summer; unexpectedly, this ties back in with the notion of generalised Hilbert object.

Tuesday, March 15, 2016

Jeff Egger,

*An update on effort locales and quasi-uniform locales, part 2*

**Abstract:**Last week, I presented a solution to the symmetry problem that so vexed me last summer---see attached notes; this week I will continue on the theme of (symmetric) effort locales as (Hermitian) modules for a Hopf quantale, and possibly also the challenge of defining a closed structure on the category of (Cauchy-complete) effort locales and effort-non-increasing maps.

Tuesday, March 22, 2016

Geoff Cruttwell (Mount Allison),

*Differential forms for tangent categories (Part I)*

**Abstract:**Tangent categories provide an axiomatic framework in which to explore differentiation and differential geometry. The canonical example of a tangent category is the category of smooth manifolds, but there are many other examples, including generalizations of the category of smooth manifolds, categories in algebraic geometry, and categories from computer science and combinatorics which involve notions of differentiation.

In a series of two talks, we will look at how to define differential forms and the exterior derivative in an arbitrary tangent category, leading to a notion of de Rham cohomology for tangent categories. The most useful definition of differential forms in tangent categories turns out to be "singular" differential forms, which look slightly different than "classical" differential forms. The two notions are equivalent in the category of smooth manifolds, but not equivalent in an arbitrary tangent category.

Moreover, a close inspection of the notion of singular differential forms leads to a further variant, "sector" differential forms. These have only made brief appearances in differential geometry literature. We will look at some interesting results for these forms which appear to be new even in the classical category of smooth manifolds.

In part I, we will review the definition of tangent categories and differential objects (the analog of vector spaces in a tangent category). Part II will look at classical, singular, and sector forms in tangent categories.

This is joint work with Rory Lucyshyn-Wright.

Tuesday, March 29, 2016

Geoff Cruttwell (Mount Allison),

*Differential forms for tangent categories (Part II)*

**Abstract:**Contuation of "Part I".

Tuesday, April 5, 2016

Bob Paré,

*Change of base for double categories*

**Abstract:**I will discuss pullback of various kinds of double functor, lax, colax etc. and their functorial properties.