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24 - Higher dimensions

from Part Two - Doing Category Theory

Published online by Cambridge University Press:  13 October 2022

Eugenia Cheng
Affiliation:
School of the Art Institute of Chicago
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Summary

In this final chapter we continue applying the principle of looking at relationships between things, giving more dimensions. We define 2-categories directly, inspired by our understanding of categories, functors, and natural transformations. We revisit the definition of category by homsets, and generalize it to give the definition of 2-category by enrichment. We revisit the definition of category by underlying graph, and generalize it to give the definition of 2-category by underlying 2-graph. We define the two types of duality for 2-categories, and discuss the appropriate notions of sameness for different dimensions of morphism in a 2-category. We define monoidal categories as 2-categories with only a 0-cell, and show the dimension shift that is analogous to the one for monoids and categories. We discuss the issue of strictness and weakness, give the unit triangle and associativity pentagon, and discuss coherence. We discuss degeneracy and the Eckmann–Hilton argument, leading to braidings. We give an introduction to how research proceeds up the dimensions, giving an overview of various approaches. The chapter becomes less formal and rigorous as we end with a taste of open research.

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Chapter
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The Joy of Abstraction
An Exploration of Math, Category Theory, and Life
, pp. 368 - 395
Publisher: Cambridge University Press
Print publication year: 2022

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  • Higher dimensions
  • Eugenia Cheng, School of the Art Institute of Chicago
  • Book: The Joy of Abstraction
  • Online publication: 13 October 2022
  • Chapter DOI: https://doi.org/10.1017/9781108769389.028
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  • Higher dimensions
  • Eugenia Cheng, School of the Art Institute of Chicago
  • Book: The Joy of Abstraction
  • Online publication: 13 October 2022
  • Chapter DOI: https://doi.org/10.1017/9781108769389.028
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Higher dimensions
  • Eugenia Cheng, School of the Art Institute of Chicago
  • Book: The Joy of Abstraction
  • Online publication: 13 October 2022
  • Chapter DOI: https://doi.org/10.1017/9781108769389.028
Available formats
×