# Spahn Monads and the Barr-Beck Theorem (Rev #2, changes)

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## 3.1 $(\infty,1)$-Categories of Endofunctors

###### 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.

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 $M$..

mapping This simplex: statement Let shall be lifted to \phi:A^0\leftarrow (\infty,1) A^1\leftarrow \dots\leftarrow A^n be a composable sequence of maps of simplicial sets. The mapping simplex of $\phi$ is denoted by $M(\phi)$.

###### Definition (composition (relative monoidal nerve) structure)

Let  M I be a simplicial category, set. let Let End^{\otimes}(M):=N_E(\Delta^{op}) f:J\to sSet and be a functor. The$\overline{End^\otimes}(M)$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:

Let now $M$ be a $(\infty,1)$-category.

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.

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 $Fun(M,M)$

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 $M(\phi)$.

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

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 $Fun(M,M)$

Revision on January 29, 2013 at 07:38:30 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.