Spahn Monads and the Barr-Beck Theorem (Rev #1, changes)

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 $s:J\to I$

2. for every nonempty subset $J^\prime\subset J$ having a maximal element $j^\prime$, a map $\tau(J^\prime):\Delta^{J}\to f(\sigma(j^\prime))$.

3. satisfying properties.

Definition (composition monoidal structure)

Let $M$ be a simplicial set. Let $End^{\otimes}(M):=N_E(\Delta^{op})$ and $\overline{End^\otimes}(M)$.

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 $Fun(M,M)\simeq End^\otimes_{[1]}(M)$.

2. 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.

Definition

Let $M$ be an $(\infty,1)$-category. Then a monad on $M$ is defined to an algebra object in $Fun(M,M)$