Tuesday, September 9, 2014
Dorette Pronk, Bicategories of Fractions Revisited
Abstract: Homs

Tuesday, September 16, 2014
Dorette Pronk, Bicategories of Fractions Revisited - Continued
Abstract: Homs

Tuesday, September 23, 2014
Dorette Pronk, Bicategories of Fractions Revisited Part 3
Abstract: Homs

Tuesday, September 30, 2014
Toby Kenney, More on Partial Sup Lattices
Abstract: In a previous seminar, we explained how partial sup lattices, i.e. posets with a partially defined operation which agrees with the order-theoretic supremum whenever it is defined, are a natural generalisation of sup-lattices, and can be used to keep track of certain suprema being structural, while others are coincidental.

In that seminar, I showed how much of the theory of sup-lattices carries over easily to partial sup lattices. In this talk, I will recap the definition of a partial sup lattice, and then focus on one particular aspect of the theory of sup-lattices, namely the downset lattice. This is an invaluable tool in the study of completely distributive sup-lattices, and in this talk, I will give the appropriate generalisation for the partial sup-lattice case, showing that much of the theory for sup-lattices extends for this case. In particular, I will show that a partial sup-lattice is completely distributive if and only if it is isomorphic to the partial sup downset construction for an idempotent relation with partial suprema.

Tuesday, October 7, 2014
Toby Kenney, Still more on Partial Sup Lattices
Abstract: Continuation of last week's talk.

Tuesday, October 14, 2014
Jeff Egger, Analytic structures overlying continuous maps

Tuesday, October 21, 2014
Mitja Mastnak, Hopf day afternoon at SMU
Program:
1:30 - M. Beattie, Introduction to Hopf algebras
2:30 – G. Garcia, Categorical quantum subgroups and Lagrangians
3:30 - V. Rodrigues, Equivariantization of abelian K-linear categories

Tuesday, October 28, 2014
Jeffrey Morton, Representing Higher Algebras: Categorification and Groupoidification Part I
Abstract: "Categorification" of some structure X is an imprecise term which means, in some sense, to give a structure Y in the setting of categories which "reduces to" X under some "decategorification" process D, so that D(Y)=X. Usually, this is the more rigorous part, and categorification becomes a lifting problem, with no guarantee of existence or uniqueness, once an operation D has been chosen. To categorify algebras, there have been different suggestions for D. This talk will relate to two of them: the algebraic categorification uses the (complexification of) the Grothendieck ring of a monoidal category; "groupoidification" uses a certain functor into Vect from the monoidal category Span(Gpd), whose objects are groupoids and whose morphisms are spans. I will show how Khovanov's algebraic categorification of the Heisenberg algebra as a certain category H built from string diagrams is related to the Baez-Dolan-Trimble groupoidification of the same algebra, by seeing the latter as part of a representation of H in the monoidal bicategory Span_2(Gpd), whose 2-morphisms are spans-of-span-maps. This is joint work with Jamie Vicary.

Tuesday, November 4, 2014
Jeffery Morton, Categorification of the Fock Monad: Categorification and Groupoidification Part II
Abstract: (Continuing from Part I) We revisit the two categorifications of the Heisenberg algebra by considering its representation on Fock space. This is the essentially unique faithful representation of the Heisenberg algebra, which appears in quantum mechanics. Abstractly, it arises from a "Fock monad", an endofunctor F on Hilb which gives the Fock space (direct sum of all symmetric tensor powers) of a given space. The structure of F as the "free commutative monoid" monad, which may be defined in any suitable monoidal category, is what gives the representation of the Heisenberg algebra on Fock space. We will see that the relation between the Khovanov and Baez-Dolan categorifications of this algebra, as described in Part I, arise from a 2-categorical analog of this fact, and a "Fock 2-monad" on the monoidal bicategory Span_2(Gpd). This is joint work with Jamie Vicary.

Tuesday, November 18, 2014
Bob Paré, Double categories and their morphisms
Abstract: I will discuss double categories as an introduction to some three dimensional generalizations, viz. intercategories.

Tuesday, November 25, 2014
Richard Wood, The waves of a total category
Abstract: The lore of cocomplete categories is in many respects not as satisfactory as that of cocomplete ordered sets. In fact, it is easy to show that a category which has all colimits is an ordered set. The situation is considerably better for totally cocomplete categories and as Max Kelly observed, most of the cocomplete categories of nature are in fact totally cocomplete --- although totally complete categories are rare in nature. (Yes, he would assert that categorical duality is not a construct of nature.)

In particular, the way below relation for complete ordered sets has a precise counterpart for totally cocomplete categories and a recently proved observation about this counterpart is the ultimate subject of the talk.

Tuesday, December 2, 2014
Richard Wood, The waves of a totally cocomplete category II
Abstract: For a totally cocomplete \K, I will show that W:\K---> \hat\K is well-defined by W(A)(K)=\set^{\hat\K}(A^*.X,\hat K)
(An element \tau of W(A)(K) is called a {\em wave from K to A} and denoted \tau:K~~~>A.) I will define \beta:WX--->1_{\hat\K} and \gamma:XW--->1_{\K} and show that \beta W= W\gamma and X\beta=\gamma X, while developing some other useful equations about waves. The point of the talk is to prove:

THEOREM For a totally cocomplete \K, the following are equivalent:
1) X has a left adjoint (that is \K is totally distributive)
2) \gamma is invertible
3) There is an adjunction <\alpha,\beta:W-|X>

Tuesday, January 6, 2015
Bob Paré Weak double categories and their morphisms
Abstract: I will define and give examples of weak double categories and their morphisms. We will see that they form a strict double category. This is an expository talk and will contain no new material.

Tuesday, January 13, 2015
Bob Paré Weak double categories and their morphisms (continued)
Abstract: I will define and give examples of weak double categories and their morphisms. We will see that they form a strict double category. This is an expository talk and will contain no new material.

Tuesday, January 20, 2015
Bob Paré, The Double Category of Double Categories
Abstract: I will introduce the cells to make a double category of double categories, lax functors and colax functors. I will then introduce weak category objects and internal lax and colax functors as well as cells.

Tuesday, February 3, 2015
Bob Paré, Weak category objects
Abstract: We will introduce the notion of weak category object in a 2-category, internal lax and colax morphisms, and show how this "explains" the cells in Dbl. Then we will examine what weak categories are in "the" 2-category of double categories. These are what we call "intercategories".

Tuesday, February 10, 2015
Nasir Sohail, Dominions, epimorphisms and amalgamation for ordered monoids
Abstract: I shall introduce the amalgamation problem for ordered monoids, and discuss its connection with dominions and epimorphisms. We know that amalgamation of monoids is subjected to `severe' restrictions after the introduction of order; I shall show that dominions and hence epis are not affected when an order is introduced on top of the algebraic structure.

Tuesday, February 24, 2015
Jeff Morton, Transformation Double Groupoids and Double Categories of Functors
Abstract: Transformation groupoids associated to group actions capture the interplay between global and local symmetries. We consider the analogous construction of a transformation double groupoid C//G associated to the action of a 2-group (categorical group) G on a category C. The two directions of morphism in the double groupoid correspond to those already in C, and new morphisms introduced to describe the action of G. We consider as an example the categorified analog of the construction of a groupoid from the moduli space of G-bundles with connection over a manifold M. This double groupoid turns out to have a second construction, analogous to the situation for ordinary groups, in which it arises as a certain double category of functors between 2-groupoids, and the two directions correspond to strict and costrict pseudonatural transformations.

Tuesday, March 3, 2015
Jeff Egger, What is a doubly involutive monoidal category?

Tuesday, March 10, 2015
Jeff Egger, What is a doubly involutive monoidal category? (Continued)

Tuesday, March 24, 2015
Geoff Cruttwell, Structure theorems for finite semigroups, groups, and categories (part I)
Abstract: Lately, I've become interested in trying to understand the structure of finite categories. To give some background to this project, in this talk I'll describe an important structure theorem for finite semigroups, the Krohn-Rhodes theorem, and its relation to the Jordan-Holder theorem for groups. In a follow-up talk, I'll discuss generalizations of these ideas to finite categories due to Wells.

Tuesday, March 31, 2015
Geoff Cruttwell, Structure theorems for finite semigroups, groups, and categories (part II)
Abstract: In this follow-up to last week's talk, I'll discuss two generalizations of "division" to finite categories, the wreath product of finite categories, and Wells' version of the Krohn-Rhodes theorem for finite categories.

Tuesday, April 7, 2015
Bob Raphael, On some reflective subcategories of the category of rings

Tuesday, April 21, 2015
Richard Wood, The interpolation lemma for the waves of a totally distributive category
Abstract: In a complete lattice, "Raney's anonymous relation", nowadays known as << is defined by k<<a iff, for all downsets S, a ≤ \/S implies k in S. Raney showed that if the complete lattice is completely distributive then k<<a implies, there exists m, with k<<m and m<<a. (He also remarked that << for the non-modular N_5 also satisfies this interpolative property, so that it does not characterize complete distibutivity.)

In this talk I will show that every wave \omega:K~~~>A in a totally distributive category factors as \psi \chi
K~~~>M~~~>A
and that the tensor \chi\ox\psi is uniquely determined. In passing from orders to categories, problems of both coherence and size occur. For example, we need to know what a composite \chi\circ\psi means to make sense of all this. The attached .pdf file for my proposed talk at CT15 provides some further context for this talk.