2,038 research outputs found

    Dualizability in Low-Dimensional Higher Category Theory

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    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 nn-groupoid of so-called `fully-dualizable' objects in any symmetric monoidal nn-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 (n≤3n \leq 3). 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 (∞,n)(\infty,n)-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

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    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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