Spahn a reading guide to HTT (Rev #3, changes)

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Appendix A.2 (model categories and their homotopy categories)

2. Fibrations of simplicial sets

2.2

2.2.5 Joyal model structure

2.3 inner fibrations and minimal inner fibrations

Every $\infty$-category is categorial equivalenct to a minimal $\infty$-category.

Corollary 2.3.2.2: $Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C)$ is a trivial fibration.

2.3.4 theory of $n$-categories

Proposition 2.3.4.19

Proposition 2.3.4.5: For a simplicial set $X$ the following statements are equivalent:

1. the unit $u:X\to N(hX)$ is an isomorphism of simplicial sets.

2. There is small category $C$ and an isomorphism of simpliial sets $X\simeq N(C)$.

3. $S$ is a 1-category.

Proposition 2.3.4.19 Let $C$ be an $\infty$-category. Let $n\ge -1$. Then the following statements are equivalent:

1. $C$ is an $n$-category.

2. For every simplicial set $K$ and every pair of maps $f,g:K\to C$ such that $f| sk^n K$ and $g|sk^n K$ are homotopic relative to $sk^{n-1}K$, we have $f=g$.

Corollary 2.3.4.8

Let $C$ be an $n$-category and let $X$ be a simplicial set. Then $Fun(X,C)$ is an $n$-category.

Proposition 2.3.4.12

Let $C$ be an $\infty$-category. Let $n\ge 1$.

1. There exists a simplicial set $h_n C$ with the following universal mapping property: $Fun(K,h_n C)=[K,C]/\sim$.

2. $h_n C$ is an $n$-category.

3. If $C$ is an $n$-category, then the natural map $\Theta:C\to h_n C$ is an isomorphism.

4. For every $n$-category $D$, composition with $\Theta$ is an isomorphism of simplicial sets $Fun(h_n C,D)\to Fun(C,D)$.

Corollary 2.3.4.19

Let $X$ be a Kan complex. Then is is equivalent to an $n$-category iff it is $n$-truncated.

2.4 cartesian fibrations

1.1 (definitions of $\infty$-categories)

$\infty$-categories as simplicial sets

$\infty$-categories as categories enriched in

1. $sSet$

2. $Top_CG$

1.2 (basic $\infty$-category theory)

1.2.3 (the homotopy category of a simplicial set)

1.2.4 (objects and morphisms in an $\infty$-category)

1.2.5 ($\infty$-groupoids)

1.2.6 (homotopy commutativity and homotopy coherence)

1.2.7 (functors between $\infty$-categories)

Proposition 1.2.7.3

1.2.10, 1.2.11, 1.2.16

4 Limits and colimits

4.1

Definition 4.1: cofinal arrow Proposition 4.1.3.1: Cofinal arrows preserve colimits

4.2

Theorem 4.2.4.1: relation of $\infty$-categorial colimits and homotopy colimits in simplicially enriched categories.

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 diagram

4.3 (Kan extensions)

4.4 Examples for limits and colimits

construction of colimits from basic diagrams

5

5.1 Presheaves

5.2

Definition 5.2.2.1

Proposition 5.2.2.6

Proposition 5.2.2.8

Proposition 5.2.2.9

Proposition 5.2.2.12

Proposition 5.2.3.5 Adjoint functors preserve (co)limits