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20 - Iterated higher categories

from PART V - HIGHER CATEGORY THEORY

Published online by Cambridge University Press:  25 October 2011

Carlos Simpson
Affiliation:
Université de Nice, Sophia Antipolis
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Summary

The conclusion of Theorem 19.3.2 matches the hypotheses we imposed that M be tractable, left proper and cartesian. Therefore, we can iterate the construction to obtain various versions of model categories for n-categories and similar objects. This process is inherent in the definitions of Tamsamani [250] and Pellissier [211]. Rezk [219] considered a modified version of the corresponding iteration of his definition, following Barwick, and Trimble's definition is also iterative (cf. Cheng [74]). Such an iteration is related to Dunn's iteration of the Segal delooping machine [97], and goes back to the well-known iterative presentation of the notion of strict n-category, see Bourn [55] for example.

Iteration leads to what might generically be called higher category theory. In the last part of the book, we explore some of the first things which can be said. The next Chapter 21 takes up one route, which is to try to generalize to the higher categorical context the large body of knowledge on usual category theory. The particular question of constructing limits and colimits of higher categories themselves is treated in Chapter 22. Chapter 23 ends the book with a look at the Breen–Baez–Dolan stabilization hypothesis. This provides a first illustration of the interaction between higher morphisms in different dimensions.

The present chapter will set up the basic context and notation, and relate higher groupoids back to homotopy theory. In what follows unless otherwise indicated, the model category denoted PC(M) will mean by definition PCReedy(M), the global model structure with Reedy cofibrations constructed in Theorem 19.3.2.

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Chapter
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Homotopy Theory of Higher Categories
From Segal Categories to n-Categories and Beyond
, pp. 455 - 479
Publisher: Cambridge University Press
Print publication year: 2011

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  • Iterated higher categories
  • Carlos Simpson, Université de Nice, Sophia Antipolis
  • Book: Homotopy Theory of Higher Categories
  • Online publication: 25 October 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511978111.021
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  • Iterated higher categories
  • Carlos Simpson, Université de Nice, Sophia Antipolis
  • Book: Homotopy Theory of Higher Categories
  • Online publication: 25 October 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511978111.021
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.

  • Iterated higher categories
  • Carlos Simpson, Université de Nice, Sophia Antipolis
  • Book: Homotopy Theory of Higher Categories
  • Online publication: 25 October 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511978111.021
Available formats
×