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

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3.1 (∞,1)(\infty,1)-Categories of Endofunctors

Definition (relative nerve)

Let II be a category, let f:J→sSetf:J\to sSet be a functor. The nerve of II relative ff denoted by N f(I)N_f(I) is defined as follows: Let JJ be a finite linear order, the a map Δ J→N f(I)\Delta^J\to N_f(I) consists of:

  1. a functor s:J→Is:J\to I

  2. for every nonempty subset J ′⊂JJ^\prime\subset J having a maximal element j ′j^\prime, a map τ(J ′):Δ J→f(σ(j ′))\tau(J^\prime):\Delta^{J}\to f(\sigma(j^\prime)).

  3. satisfying properties.

For an (∞,1)(\infty,1)-category MM the (∞,1)(\infty,1)-category Fun(M,M)Fun(M,M) is a monoid object in the category sSetsSet and MM is endowed with a (11-categorial) left action of Fun(M,M)Fun(M,M) and this action is universal among left actions on MM..

mapping This simplex: statement Let shall be lifted toϕ(:∞A 0,←1A 1)←…←A n \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 ϕ\phi is denoted by M(ϕ)M(\phi).

Definition (composition (relative monoidal nerve) structure)

Let M I M I be a simplicial category, set. let LetEnd ⊗f(M):=JN E→(sSetΔ op) End^{\otimes}(M):=N_E(\Delta^{op}) f:J\to sSet and be a functor. TheEnd ⊗¯(M)\overline{End^\otimes}(M)nerve of II relative ff . denoted byN f(I)N_f(I) is defined as follows: Let JJ be a finite linear order, the a map Δ J→N f(I)\Delta^J\to N_f(I) consists of:

Let now MM be a (∞,1)(\infty,1)-category.

  1. a functor s:J→Is:J\to I

  2. for every nonempty subset J ′⊂JJ^\prime\subset J having a maximal element j ′j^\prime, a map τ(J ′):Δ J→f(σ(j ′))\tau(J^\prime):\Delta^{J}\to f(\sigma(j^\prime)).

  3. satisfying properties.

  1. The map p:End ⊗(M)→N(Δ) opp:End^{\otimes}(M)\to N(\Delta)^{op} determines a monoidal structure on the (∞,1)(\infty,1)-category Fun(M,M)≃End [1] ⊗(M)Fun(M,M)\simeq End^\otimes_{[1]}(M).

  2. The map q:End ⊗¯→End ⊗(M)q:\overline{End^\otimes}\to End^\otimes(M) exhibits M≃End [0] ⊗¯(M)M\simeq \overline{End^\otimes_{[0]}}(M) as left tensored over Fun(M,M)Fun(M,M).

This monoidal structure on Fun(M,M)Fun(M,M) is called the composition monoidal structure.
Definition

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

mapping simplex: Let ϕ:A 0←A 1←…←A n\phi:A^0\leftarrow A^1\leftarrow \dots\leftarrow A^n be a composable sequence of maps of simplicial sets. The mapping simplex of ϕ\phi is denoted by M(ϕ)M(\phi).
Definition (composition monoidal structure)

Let MM be a simplicial set. Let End ⊗(M):=N E(Δ op)End^{\otimes}(M):=N_E(\Delta^{op}) and End ⊗¯(M):=N E¯(Δ op)\overline{End^\otimes}(M):=N_{\overline E}(\Delta^{op}).

Let now MM be a (∞,1)(\infty,1)-category.

  1. The map p:End ⊗(M)→N(Δ) opp:End^{\otimes}(M)\to N(\Delta)^{op} determines a monoidal structure on the (∞,1)(\infty,1)-category Fun(M,M)≃End [1] ⊗(M)Fun(M,M)\simeq End^\otimes_{[1]}(M).

  2. The map q:End ⊗¯→End ⊗(M)q:\overline{End^\otimes}\to End^\otimes(M) exhibits M≃End [0] ⊗¯(M)M\simeq \overline{End^\otimes_{[0]}}(M) as left tensored over Fun(M,M)Fun(M,M).

This monoidal structure on Fun(M,M)Fun(M,M) is called the composition monoidal structure.
Definition

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

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