## 2.0 +-- {: .un_defn} ###### Definition A morphism of simplicial sets is called 1. a *Kan fibration* if it has the right lifting property with respect to every horn inclusion. 1. a *trivial fibration* if it has the right lifting property with respect to every boundary inclusion $\partial \Delta[n]\hookrightarrow \Delta[n]$. 1. a *left fibration* if it has the right lifting property with respect to every horn inclusion except the right outer one. 1. a *left fibration* if it has the right lifting property with respect to every horn inclusion except the left outer one. 1. a *left fibration* if it has the right lifting property with respect to every inner horn inclusion. 1. *left anodyne* if it has the left lifing property with respect to every left fibration. 1. *right anodyne* if it has the left lifing property with respect to every right fibration. 1. *inner anodyne* if it has the left lifing property with respect to every inner fibration. 1. *minimal fibration* roughly said, when the morphism is determined by its values on the boundaries. 1. *cartesian fibration* 1. *cocartesian fibration* 1. *categorical fibration* =-- +-- {: .num_remark} ###### Remark We have the following intuition in regard to these types of fibrations 1. Right fibrations are the $\infty$-categorical analog of fibrations in groupoids. 1. Left fibrations are the $\infty$-categorical analog of cofibrations in groupoids. 1. Cartesian fibrations are the $\infty$-categorical analog of fibrations (not necessarily in groupoids). =-- #### 2.2.5 Joyal model structure ### 2.3 Inner fibrations and minimal inner fibrations Corollary 2.3.2.2: $Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C)$ is a trivial fibration. Every $\infty$-category is categorial equivalenct to a minimal $\infty$-category. #### 2.3.3. Minimal inner fibrations +-- {: .un_defn} ###### Definition 2.3.3.1 Let $p : X \to S$ be an inner fibration of simplicial sets. $p$ is called *minimal inner fibration* if $f = f^\prime$ for every pair of maps $f , f ^\prime : \Delta[n] \to X$ which are homotopic relative to $\partial \Delta[n]$ over $S$ . An $\infty$-category $C$ is called *minimal $\infty$-category* if $C\to *$ is minimal. =-- (...) Every $\infty$-category is equivalent to a minimal $\infty$-category. #### 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. $Top_CG$