Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories

Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories

Science & Math
ISBN: 3030056732Format: PDFEdition: 1st ed. 2019 editionDate: June 4, 2019Pages: 343 pagesLanguage: English

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Book Description

This monograph presents a new model of mathematical structures called weak n-categories. These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic.
While strict n-categories are easily defined in terms associative and unital composition operations they are of limited use in applications, which often call for weakened variants of these laws. The author proposes a new approach to this weakening, whose generality arises not from a weakening of such laws but from the very geometric structure of its cells; a geometry dubbed weak globularity. The new model, called weakly globular n-fold categories, is one of the simplest known algebraic structures yielding a model of weak n-categories. The central result is the equivalence of this model to one of the existing models, due to Tamsamani and further studied by Simpson. This theory has intended applications to homotopy theory, mathematical physics and to long-standing open questions in category theory.
As the theory is described in elementary terms and the book is largely self-contained, it is accessible to beginning graduate students and to mathematicians from a wide range of disciplines well beyond higher category theory. The new model makes a transparent connection between higher category theory and homotopy theory, rendering it particularly suitable for category theorists and algebraic topologists. Although the results are complex, readers are guided with an intuitive explanation before each concept is introduced, and with diagrams showing the interconnections between the main ideas and results.

Content

Part I Higher Categories: Introduction and Background
1. An Introduction to Higher Categories
2. Multi-Simplicial Techniques
3. An Introduction to the Three Segal-TypeModels
4. Techniques from 2-Category Theory

Part II The Three Segal-Type Models and Segalic Pseudo-Functors
5. Homotopically Discrete n-Fold Categories
6. The Definition of the Three Segal-Type Models
7. Properties of the Segal-Type Models
8. Pseudo-Functors Modelling Higher Structures

Part III Rigidification of Weakly Globular Tamsamani n-Categories
9. ApproximatingWeakly Globular Tamsamani n-Categories by Simpler Ones
10. RigidifyingWeakly Globular Tamsamani n-Categories

Part IV Weakly Globular n-Fold Categories as a Model of Weak n-Categories
11. Functoriality of Homotopically Discrete Objects
12. Weakly Globular n-Fold Categories as a Model of Weak n-Categories
13. Conclusions and Further Directions

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Simplicial Methods for Higher Categories: Segal-type Models of Weak n-Categories
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