## 3.1 $(\infty,1)$-Categories of Endofunctors For an $(\infty,1)$-category $M$ the $(\infty,1)$-category $Fun(M,M)$ is a monoid object in the category $sSet$ and $M$ is endowed with a ($1$-categorial) left action of $Fun(M,M)$ and this action is universal among left actions on $M$.. This statement shall be lifted to $(\infty,1)$. +-- {: .un_defn} ###### Definition (relative nerve) Let $I$ be a category, let $f:J\to sSet$ be a functor. The *nerve of $I$ relative $f$* denoted by $N_f(I)$ is defined as follows: Let $J$ be a finite linear order, the a map $\Delta^J\to N_f(I)$ consists of: 1. a functor $s:J\to I$ 1. for every nonempty subset $J^\prime\subset J$ having a maximal element $j^\prime$, a map $\tau(J^\prime):\Delta^{J}\to f(\sigma(j^\prime))$. 1. satisfying properties. =-- mapping simplex: Let $\phi:A^0\leftarrow A^1\leftarrow \dots\leftarrow A^n$ be a composable sequence of maps of simplicial sets. The *mapping simplex of $\phi$* is denoted by $M(\phi)$. +-- {: .un_defn} ###### Definition (composition monoidal structure) Let $M$ be a simplicial set. Let $End^{\otimes}(M):=N_E(\Delta^{op})$ and $\overline{End^\otimes}(M):=N_{\overline E}(\Delta^{op})$. Let now $M$ be a $(\infty,1)$-category. 1. The map $p:End^{\otimes}(M)\to N(\Delta)^{op}$ determines a monoidal structure on the $(\infty,1)$-category $Fun(M,M)\simeq End^\otimes_{[1]}(M)$. 1. The map $q:\overline{End^\otimes}\to End^\otimes(M)$ exhibits $M\simeq \overline{End^\otimes_{[0]}}(M)$ as left tensored over $Fun(M,M)$. This monoidal structure on $Fun(M,M)$ is called the *composition monoidal structure*. =-- +-- {: .un_defn} ###### Definition Let $M$ be an $(\infty,1)$-category. Then a *monad on $M$* is defined to an algebra object in $Fun(M,M)$ =--