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Contents

The reading strategy outlined here is approximately the following:

• start Start with appendix A.2.

• continue Continue with the overview chapter 2. 1.

• omit Chapter chapter 2 3. 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.

• the Omit rest of the book is concerned with constructions which in most cases are proposed in chapter 2. 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)

  • the Yoneda lemma and presheaves

  • limits and colimits

  • ind-objects

  • adjoint functors

  • $\infty$-topoi

A.2 Model categories

HTT, A.2 model categories

1. An overview of higher category theory

HTT, 1. an overview of higher category theory

4 Limits and colimits

Proposition 4.2.4.4 (and Corollary 4.2.4.7)in a simplicial model category every homotopy coherent diagram is equivalent to a commutative diagramHTT, 6. infinity-topoi?