## 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 +-- {: .un_prop} ###### 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. 1. There is small category $C$ and an isomorphism of simpliial sets $X\simeq N(C)$. 1. $S$ is a 1-category. =-- +-- {: .un_prop} ###### 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. 1. 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 $f=g$. =-- +-- {: .un_corollary} ###### 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. =-- +-- {: .un_prop} ###### 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: $Fun(K,h_n C)=[K,C]/\sim$. 1. $h_n C$ is an $n$-category. 1. If $C$ is an $n$-category, then the natural map $\Theta:C\to h_n C$ is an isomorphism. 1. For every $n$-category $D$, composition with $\Theta$ is an isomorphism of simplicial sets $Fun(h_n C,D)\to Fun(C,D)$. =-- +-- {: .un_corollary} ###### Corollary 2.3.4.19 Let $X$ be a [[nLab: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. $sSet$ 1. $Top_CG$ ## 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