We review the general notion of categorification in order to study quantization -- in the sense of deformation quantization -- and Feynman integrals from the viewpoint of category theory. The fairly abstract setting that we propose leads to a rather down to earth approach to these often elusive notions.

This talk will introduce the construction "2-linearization", an extension of the "groupoidification" program of Baez and Dolan which seeks to interpret constructions in linear algebra in terms of groupoids and spans of groupoids. The 2-linearization construction finds this as a special case of a 2-functor $\Lambda :\mathrm{Span}(\mathrm{Gpd})\to 2\mathrm{Vect}$, where $2\mathrm{Vect}$ is the 2-category of Kapranov-Voevodsky 2-vector spaces. This 2-functor is constructed in terms of pairs of ambi-adjoint functors associated to each groupoid homomorphism, the "push" and "pull" operations, closely related to restriction and induction maps in representation theory, Grothendieck's 6-operation framework for sheaves, among other examples. We will begin with the discrete case, and consider generalizations to smooth groupoids. Finally we will consider some applications.

M. Khovanov and L. Rozansky constructed a homology for a link diagram whose Euler characteristic is the quantum link invariant associated to the quantum group $\mathrm{Uq}({\mathrm{sl}}_{n})$ and its vector representation ${V}_{n}$ by using matrix factorizations. In my thesis, I study a generalization of the Khovanov-Rozansky homology for the quantum link invariant associated to $\mathrm{Uq}({\mathrm{sl}}_{n})$ and its fundamental representations $\wedge {V}_{n}$. In this talk, I will define a new link invariant derived from the generalization of Khovanov-Rozansky homology.

We review the general notion of categorification in order to study quantization -- in the sense of deformation quantization -- and Feynman integrals from the viewpoint of category theory. The fairly abstract setting that we propose leads to a rather down to earth approach to these often elusive notions.

We generalize the BF theory action to the case of a general Lie crossed module $(H\to G)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie algebras of $G$ and $H$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short-complexes of vector spaces. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $H$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. We also construct a two-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere. http://arxiv.org/abs/1006.0903

Support: FCT, CAMGSD, New Geometry and Topology.Support: FCT, CAMGSD, New Geometry and Topology.

We consider a monoidal category consisting of cubic graphs with free ends as morphisms between finite sets of points on a line (in the same sense as the category of tangles). We give a linear representation of this category that codifies the number of edge 3-colorings of cubic graphs. Some developments and conjectures will be presented.

Rouquier categorified braid groups, in the sense that he associated to each braid a complex of Soergel bimodules such that complexes associated to isotopic braids are homotopy equivalent. I will start with recalling Rouquier's construction and then extend this result to singular braids and to virtual braids.

This is meant to be a (rather incomplete) survey talk of the rapidly developing field of categorification. I will only concentrate on two of the various now existing approaches: the one using (singular) Soergel bimodules, due to Soergel (1992) and Williamson (2008), and the one using diagrams due to Khovanov and Lauda (2008) and Elias and Khovanov (2009). My intention is to explain the relation between the two approaches, as was worked out by Stosic, Vaz and myself.

The Baez-Dolan groupoidification program describes linear-algebraic structures as reflections of those found in a category whose objects are groupoids, and whose morphisms are spans of groupoids. An extension of this program represents those structures in a 2-category of 2-vector spaces. Since quantum physics relies heavily on the category of Hilbert spaces, it has been possible to describe certain toy physical systems in this setting. The talk will describe these in terms of groupoids and spans with intrinsic combinatorial and geometric interest, and discuss how the 2-categorical extension applies in these models, including exotic statistics, and applications to 3D quantum gravity.

Measure theory is the study of measures and their siamese twins, integrals. Its central position in analysis is partly explained because it provides us with large families of Banach spaces. In this seminar I will try to explain the why, the what and the how of categorifying measure theory. It will consist mostly in laying the groundwork so as to be able to explain the construction of the category of \"categorified integrable functionsänd the integral functor. Time permitting, I will also give a categorified Radon- Nikodym theorem. In a second, future seminar, I will give categorified versions of other basic theorems of measure theory (e.g. Fubini and Fubini-Tonelli on the equality of iterated integrals) and explain some new phenomena peculiar to the categorified setting and with no counterpart in ordinary measure theory.

One can obtain simple models for abstract field theories by looking at closed forms defined on the total space of a fibration. For instance, in the case of hamiltonian fibrations, this is how Weinstein recovered Sternberg's minimal coupling, yielding a geometrical context for classical Yang-Mills theories. More generally, one can introduce 2-plectic fibrations and interpret the equations for coupling in terms of higher analogues of connections.

In this talk I plan to fill in some details of what I sketched in my previous talk. In particular I want to explain how one can obtain Khovanov and Lauda\'s calculus from bimodule maps corresponding to MOY-movies. In the end I will sketch the extended sl2 calculus for divided powers. References: 1) A. Lauda A categorification of quantum sl2, arXiv:0803.3652. 2) M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I-III, arXiv:0803.4121, arXiv:0804.2080, arXiv:0807.3250. 3) M. Khovanov, A. Lauda, M. Mackaay, M. Stosic, Extended graphical calculus for categorified quantum sl2, arXiv:1006.2866. 4) M. Mackaay, M. Stosic, P. Vaz, A diagrammatic categorification of the q-Schur algebra, arXiv:1008.1348.

It is well known that connections on a principal G-bundle over a manifold M can be represented by group homomorphisms LM->G, where LM is the loop group of M. Similarly, 2-connections can be seen as 2-functors from a 2-groupoid of points, paths and bigons in M to a 2-group G. In passing to dimension 3, that is, considering tri-connections, 3-groupoids are too rigid, instead one needs to consider Gray-groupoids, their morphisms and transformations of higher dimension. We will describe a groupoid structure for certain functors between a pair of Gray-categories and certain transformations between them.

This lecture will be a continuation of my October lecture on a program to categorify measure theory. After a quick reminder of some of the basic concepts like the finite Grothendieck topology and cosheaves, and the basic theorem on the (bi)representability of the the cosheaf category, I will try to demonstrate that all the work gone into setting up the machinery was not in vain by first, giving categorified versions of some basic theorems of measure theory (e.g. Fubini and Fubini-Tonelli) and second, by explaining some of the rich structure underlying categorified measure theory, a blend of analysis, algebraic geometry and topos theory.