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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
A Cognitive Mind-map Framework to Foster Trust
The explorative mind-map is a dynamic framework, that emerges automatically
from the input, it gets. It is unlike a verificative modeling system where
existing (human) thoughts are placed and connected together. In this regard,
explorative mind-maps change their size continuously, being adaptive with
connectionist cells inside; mind-maps process data input incrementally and
offer lots of possibilities to interact with the user through an appropriate
communication interface. With respect to a cognitive motivated situation like a
conversation between partners, mind-maps become interesting as they are able to
process stimulating signals whenever they occur. If these signals are close to
an own understanding of the world, then the conversational partner becomes
automatically more trustful than if the signals do not or less match the own
knowledge scheme. In this (position) paper, we therefore motivate explorative
mind-maps as a cognitive engine and propose these as a decision support engine
to foster trust.Comment: 5 pages, 4 Figures, Extended Version, presented at the 5th
International Conference on Natural Computation, 200
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