Tuesday, September 10, 2013
Geoff Cruttwell, Tangent categories, vector bundles, and connections I
Abstract: The tangent bundle is a fundamental object in differential geometry. It consists of the space of all "tangent vectors" associated to a smooth space. Though it is fundamental, it is also difficult to describe precisely, and takes some work to construct.

An alternative approach to understanding the tangent bundle was proposed in 1984 by Rosicky. Rather than give an explicit construction of the tangent bundle, one can instead describe its abstract structures and properties and work with them. (This is very similar to how we rigourously deal with the real numbers: rather than work with an explicit construction of real numbers such as that of Dedekind or Cauchy, we instead work with a complete ordered field). We call Rosicky's axiomatization a "tangent category". It consists of a category with an endofunctor, several associated natural transformations, and some limits. As a categorical structure, it is a common generalization of the Cartesian differential categories of Blute, Cockett and Seely and the synthetic differential geometry of Kock and Lawvere.

In a series of two talks, I'll describe tangent categories and some of their theory: that is, how to define and prove results from differential geometry in an arbitrary tangent category. In particular, I'll focus on how to define vector bundles and connections in this abstract setting. The abstract definitions shed light on the standard definitions, and in some cases simplify them.

No prior knowledge of differential geometry is required. This is joint work with Robin Cockett.

Tuesday, September 17, 2013
Geoff Cruttwell, Tangent categories, vector bundles, and connections II
Abstract: Continuation of last week's talk.

Tuesday, September 24, 2013
Dorette Pronk, Mapping orbifolds

Tuesday, October 1, 2013
Toby Kenney, Distributing over non-existant suprema
Abstract: In previous work with Richard Wood, we investigated using the +- monad as a substitute for suprema in cases where a poset does not have suprema. This allowed us to obtain a definition of complete distributivity for posets which were not lattices. In this talk, I extend this study, giving more definitions of complete distributivity, and weaker concepts such as frames and distributive lattices, without requiring them to have joins.
Tuesday, October 8, 2013
Richard Wood, Big Categories and Big Limits --- an introduction to cototal categories
Abstract: It is well known that the size of categories presents some difficult problems. At the outset, it is best for students to ignore these problems and learn first the calculus of adjoints, Kan extensions, and limits. But soon it is clear that not all categories can have all limits.

The orthodoxy of the subject seems to suggest that most of the categories of "nature" --- the categories of useful mathematical structures and the arrows that suitably compare, or even preserve, such structures --- are locally small, have all small limits, all small colimits, and that the 2-categories of categories that arise from these endeavours are distinguished, from each other, by equational-like relationships between said limits and said colimits.

It is not often acknowledged that many of the categories of nature have some limits and some colimits that are very large. The point of the work, by the speaker, Francisco Marmolejo, and Bob Rosebrugh that this talk attempts to introduce, is to find a framework that allows for "distributivities" between the "large limits" and the "large colimits" that is not accounted for by the orthodoxy.

Tuesday, October 15, 2013
Richard Wood, Big Categories and Big Colimits --- an introduction to total categories
Abstract: In last week's rather cursory introduction to totally complete categories we showed that, for any locally small category \cal K, the functor category \hat{\cal K}=CAT(\cal K^op,set) is totally complete. As the reader no doubt suspects, a category is said to be {\em totally cocomplete} (often abbreviated to {\em total}) if its dual is totally complete. We will begin this week by showing that, for any locally small \cal K, \hat{\cal K} is totally cocomplete. Just as conical (ordinary) colimits in set are computed very differently from their conical limit counterparts, so it is that the functor witnessing total cocompleteness of \hat{\cal K} is very different from that witnessing total completeness of \hat{\cal K}. Pleasingly, it is explicitly set-valued.

Also pleasing is the fact that while total completeness is rare in "nature", total cocompleteness is very common. In addition to powers of set, only categories of sheaves, the categories of small topological spaces and of small abelian groups come readily to the mind of the speaker. All these examples and categories of universal algebras are total and totality is stable under construction of categories of coalgebras for suitable comonads.

We will recall the usual definitions of both sides of totality as adjoints to the Yoneda functors, for locally small categories, and introduce new definitions in terms of the Yoneda profunctors, that exist for {\em all} categories. This is joint work with Francisco Marmolejo, and Bob Rosebrugh.

Tuesday, October 22, 2013
Margaret Beattie, Quantum lines for dual quasi-bialgebras
Abstract: A dual quasi-bialgebra $(H, \omega)$ is a coalgebra $H$ with a unit $u$ and a map $m: H \otimes H \rightarrow H$, called multiplication, together with a $3$-cocycle $\omega$ such that $m(H \otimes m) \ast u\omega = u \omega \ast m(m \otimes H)$. Thus multiplication need not be associative. For example, let $H$ be an ordinary bialgebra and twist the multiplication in $H$ by a unitary, convolution invertible map $v: H \otimes H \rightarrow \Bbbk$, where $\Bbbk$ is the base field. Then $H^v$ may no longer be an ordinary bialgebra but is a dual quasi-bialgebra.

We study bosonizations as defined by Ardizzoni and Pavarin for dual quasi-bialgebras, in particular, bosonizations of the form $R \# H$ where $H$ is a dual quasi-bialgebra and $R$ is a bialgebra in the category $_H^H \mathcal{YD}$ of left-left Yetter-Drinfeld modules generated as an algebra by a one-dimensional vector space.

Our construction is general but our examples focus on the case where $(H, \omega) = (\Bbbk C_n, \omega)$ or $H$ is a bosonization $R \# \Bbbk C_n$ and thus relate to the duals of those in the work of Etingof and Gelaki, and of Angiono.

Tuesday, October 29, 2013
Jeff Egger, Groupoids in non-commutative geometry, part 1: revisiting Mackey's "virtual groups"
Abstract: Groupoids, of one sort or another, generate most (if not all) of the interesting examples in non-commutative geometry (NCG). Mackey's theory of "virtual groups", now seen as a forerunner of NCG, is explicitly based in terms of (measured) groupoids, so it seems appropriate to begin my study of how much of NCG can be understood directly in terms of groupoids, and their classifying toposes.

Tuesday, November 5, 2013
Jeff Egger, Groupoids in non-commutative geometry, part 1.5: Continuing to revisit Mackey's "virtual groups"
Abstract: I will briefly recap the main definitions and examples from my previous talk, and proceed to Mackey's definition of "virtual homomorphism" between "virtual subgroups" of actual groups. This leads directly to measure groupoids.

Tuesday, November 12, 2013
Bob Rosebrugh, Symmetric lenses and spans
Abstract: What we now call "asymmetric lenses" were introduced to provide a strategy for the view update problem: given model domains X and Y, a "get" operation g : X --> Y, and update operations u : Y --> Y, find a lift of the u to X. We have studied several kinds of asymmetric lenses depending on what structure is assumed for the model domains and what is required of the lift operation. A related problem is the following: suppose model states x,y from model domains X,Y are known to be "synchronized" by some (external) information c. For an update to one of the model states, say from x to x', how should it propagated to an update of the other state, say from y to y', with the updated model states x',y' re-synchronized by some c'? Recently several kinds of what are called "symmetric lenses" address the problem. Asymmetric lenses and (equivalence classes of) symmetric lenses of the various kinds can be viewed as arrows of corresponding categories. Symmetric lenses of each sort arise as spans of asymmetric lenses of the related sort. Joint work with Michael Johnson.

Tuesday, November 19, 2013
Toby Kenney, Spin homomorphisms and classification of STB posets
Abstract: Previously, I talked about how to define complete distributivity for arbitrary posets (termed STB). For completely distributive lattices, there is an important result of Raney, that any completely distributive lattice is a complete-lattice quotient of a complete sublattice of a power set lattice. In this talk, I will extend this result to STB posets, on the way addressing interesting questions about STB posets, such as what is the equivalent of a complete lattice homomorphism.

Tuesday, November 26, 2013
Bob Paré, Semidirect and bicrossed products of groups and things
Abstract: In this largly expository talk I'll discuss the semidirect product of groups and it's generalization, bicrossed product used by the Hopf algebra theorists. I'll try to put these constructions in a more categorical context, i.e. something I understand better.

Tuesday, December 3, 2013
Bob Paré, Yetter- Drinfeld modules
Abstract: I take up the theme from last week's talk, but approach it from the opposite direction. I will discuss monoidal categories, emphasizing symmetric and braided ones. Then introduce the centre construction and its relation to Yetter-Drinfeld modules. The talk will be mostly expository.

Tuesday, January 7, 2014
Jeff Egger, Enriched dagger categories of Hilbert spaces, part 1
Abstract: We review Hilbert spaces, Banach spaces, and perhaps even operator spaces, if time permits. By this, I mean to say that I will discuss the various involutive monoidal structures of Ban (the category of Banach spaces and linear contractions) and Oper (the category of operator spaces and linear complete contractions), and describe various Ban- and Oper-enriched dagger categories whose objects are Hilbert spaces. Note: This is not a continuation of last term's groupoids talk. Notes

Tuesday, January 14, 2014
Jeff Egger, Enriched dagger categories of Hilbert spaces, part 2
Abstract: We continue last week's talk with an emphasis on the category of operator spaces. Notes

Tuesday, January 21, 2014
Jeff Egger, Enriched dagger categories of Hilbert spaces, part 3
Abstract: In my last two talks*, I discussed Ban- and Oper-enriched categories of Hilbert spaces; in this talk, I introduce an Oper^op-enriched category of Hilbert spaces, with an eye toward understanding unitary representations of arbitrary (that is, not necessarily discrete) locally compact groups.

Tuesday, February 25, 2014
Gabor Lukacs, Bornologies in topology, topological algebras, and their dualities I

Tuesday, March 4, 2014
Gabor Lukacs, Bornologies in topology, topological algebras, and their dualities II

Tuesday, March 11, 2014
Gabor Lukacs, Bornologies in topology, topological algebras, and their dualities III

Tuesday, March 18, 2014
Bob Paré, Duoidal Categories
Abstract: This will be an expository talk on Duoidal categories based on the work of Aguiar and Mahajan (who call them 2-monoidal categories) and Booker and Street. These are categories equipped with two related monoidal structures and generalize braided monoidal categories.

Tuesday, April 1, 2014
Bob Paré, Duoidal Categories (continued)

Tuesday, April 29, 2014
Toby Kenney, Completely Distributive Partial Lattice Complete Partial Orders
Abstract: The study of sup-lattices teaches us the important distinction between the algebraic part of the structure (in this case suprema) and the coincidental part of the structure (in this case infima). While a sup-lattice happens to have all infima, only the suprema are part of the algebraic structure.

Extending this idea, we look at posets that happen to have all suprema (and therefore all infima), but we will only declare some of them to be part of the algebraic structure (which we will call joins). We find that a lot of the theory of complete distributivity for sup-lattices can extend to this context. There are a lot of natural examples of completely distributive partial lattice complete partial orders, including for example, the lattice of all equivalence relations on a set X, and the lattice of all subgroups of a group G. In both cases we define the join operation as union. This is a partial operation, because for example, the union of subgroups of a group is not necessarily a subgroup. However, sometimes it is, and keeping track of this can help with topics such as the inclusion-exclusion principle.

Another motivation for the study of sup-lattices is as a simplified model for the study of presheaf categories. The construction of downsets is a form of the Yoneda embedding. In this context, partial lattices can be viewed as a simplified model for the study of sheaf categories. We can make a similar downset construction, which is a form of sheaf category.

Tuesday, May 6, 2014
Jeff Egger, Preliminaries to “the social life of generalised Hilbert objects”
Notes: Click here

Tuesday, May 13, 2014
Jeff Egger, On the social life of generalised Hilbert objects
Abstract: Involutive monoidal categories (IMC's) provide a nice framework to discuss generalised Hilbert objects (GHO's); in this talk I will show how to pass GHO's from one IMC to another---which is trickier than it ought to be!