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Contents

This entry is about the text

• Jacob Lurie, higher topos theory, arXiv:math/0608040v4

This book has prerequisites: category theory (see also the exposition in appendix A.1), in particular realization and nerve.

The reading strategy outlined here is approximately the following:

• Start with appendix A.2.

• Continue with the overview chapter 1.

• Chapter 2 developes the theory of fibrations of simplicial sets.The aim of this are mainly three different concerns:

  • Establishing the $\infty$-Grothendieck construction: The type of fibrations accomplishing this are left/right fibrations (aka. fibrations in groupoids) and cartesian fibrations (aka. Grothendieck fibrations).

  • Preparing the Joyal model structure: This is a foundational topic; the fibrant objects of this model structure are precisely $/infty$-categories. The technical vehicle for this are anodyne maps.

  • Provide a foundations for a theory of $n$-categories, for any $n\le\infty$. For the well definedness of this notion minimal fibrations (a special kind of inner fibrations) are introduced.

• Omit chapter 3.

• The rest of the book is concerned with constructions which in most cases are proposed in chapter 2. So concentrate on the following important topics:

  • the Grothendieck construction (already in chapter 2) 2),Grothendieck construction in HTT

  • the Yoneda lemma and presheaves presheaves,Yoneda lemma in HTT?

  • limits and colimits

  • ind-objects

  • adjoint functors

  • $\infty$-topoi

A.2 Model categories

HTT, A.2 model categories

A.3 Simplicial categories

HTT, A.3 simplicial categories

1. An overview of higher category theory

HTT, 1. an overview of higher category theory

2. Fibrations of simplicial sets

HTT, fibrations of simplicial sets

4 Limits and colimits

HTT, 4. limits and colimits

5 Presentable and accessible $\infty$-categories

HTT, 5. presentable and accessible infinity-categories

6. $\infty$-Topoi

HTT, 6. infinity-topoi?