11 research outputs found
Dualizability in Low-Dimensional Higher Category Theory
These lecture notes form an expanded account of a course given at the Summer
School on Topology and Field Theories held at the Center for Mathematics at the
University of Notre Dame, Indiana during the Summer of 2012. A similar lecture
series was given in Hamburg in January 2013. The lecture notes are divided into
two parts.
The first part, consisting of the bulk of these notes, provides an expository
account of the author's joint work with Christopher Douglas and Noah Snyder on
dualizability in low-dimensional higher categories and the connection to
low-dimensional topology. The cobordism hypothesis provides bridge between
topology and algebra, establishing important connections between these two
fields. One example of this is the prediction that the -groupoid of
so-called `fully-dualizable' objects in any symmetric monoidal -category
inherits an O(n)-action. However the proof of the cobordism hypothesis outlined
by Lurie is elaborate and inductive. Many consequences of the cobordism
hypothesis, such as the precise form of this O(n)-action, remain mysterious.
The aim of these lectures is to explain how this O(n)-action emerges in a range
of low category numbers ().
The second part of these lecture notes focuses on the author's joint work
with Clark Barwick on the Unicity Theorem, as presented in arXiv:1112.0040.
This theorem and the accompanying machinery provide an axiomatization of the
theory of -categories and several tools for verifying these axioms.
The aim of this portion of the lectures is to provide an introduction to this
material.Comment: 65 pages, 8 figures. Lecture Note
The balanced tensor product of module categories
The balanced tensor product M (x)_A N of two modules over an algebra A is the
vector space corepresenting A-balanced bilinear maps out of the product M x N.
The balanced tensor product M [x]_C N of two module categories over a monoidal
linear category C is the linear category corepresenting C-balanced right-exact
bilinear functors out of the product category M x N. We show that the balanced
tensor product can be realized as a category of bimodule objects in C, provided
the monoidal linear category is finite and rigid.Comment: 19 pages; v3 is author-final versio
Modular categories as representations of the 3-dimensional bordism 2-category
We show that once-extended anomalous 3-dimensional topological quantum field
theories valued in the 2-category of k-linear categories are in canonical
bijection with modular tensor categories equipped with a square root of the
global dimension in each factor.Comment: 71 page
Extended 3-dimensional bordism as the theory of modular objects
A modular object in a symmetric monoidal bicategory is a Frobenius algebra
object whose product and coproduct are biadjoint, equipped with a braided
structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and
modularity conditions. We prove that the oriented 3-dimensional bordism
bicategory of 1-, 2-, and 3-manifolds is the free symmetric monoidal bicategory
on a single anomaly-free modular object.Comment: 64 page
From the cobordism hypothesis to higher Morse theory
Non UBCUnreviewedAuthor affiliation: Max Planck Institute for MathematicsFacult