Spahn
Monads and the Barr-Beck Theorem (Rev #7, changes)

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

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.

Recall how the (,1)(\infty,1)-Grothendieck construction works in the following example: Let II be a 11-category, let f:N(I)(,1)Catf:N(I)\to (\infty,1)Cat be a diagram. We obtain the desired cartesian fibration XN(I)X\to N(I) by first replacing ff by a simplicial functor F:N coh(I) opsSet +F:N_{coh}(I)^{op}\to sSet^+ where N cohN_{coh} denotes the homotopy coherent nerve functor and sSet +sSet^+ denotes the category of marked simplicial sets. FF is a weakly fibrant object of (sSet +) N coh(I) op(sSet^+)^{N_{coh}(I)^{op}}. Applying the unstraightening functor? Un N(J) +Un^+_{N(J)} we obtain a fibrant object sSet +/N(I)sSet^+/N(I) which we identify with the desired cartesian fibration p:XN(I)p:X\to N(I).

This statement shall be lifted to (,1)(\infty,1).

Preparation

Definition (C C^{\otimes}, monoidal (,1)(\infty,1)-category)

Let CC be a symmetric monoidal category with tensor \otimes. The category C C^\otimes consists of the following data:

(1) Objects are finite sequences of CC-objects [C 1,,C n][C_1,\dots,C_n].

(2) Morphisms f:[C 1,,C n][C 1 ,C m ]f:[C_1,\dots,C_n]\to [C_1^',\dots C_m^'] are pairs

(a S:S{1,,m},(f j: a(i)=jC iC j ) 1jm)(a_S:S\to\{1,\dots,m\}, (f_j:\otimes_{a(i)=j}C_i\to C_j^')_{1\le j\le m})

where S{1,,n}S\subseteq \{1,\dots,n\} is a subset (or rather isomorphic in SetSet to a subset) .

(3) Composition of f:=(a S,(f i) i):[C 1,,C n][C 1 ,C m ]f:=(a_S,(f_i)_i):[C_1,\dots,C_n]\to [C_1^',\dots C_m^'] and g:=(b T,(g j) j):[C 1 ,,C m ][C 1 ,C l ]g:=(b_T,(g_j)_j):[C_1^',\dots,C_m^']\to [C_1^{''},\dots C_l^{''}] is defined to be

gf:=(c a 1(T), ba)(i)=kC i b(j)=k a(i)=jC i b(j)=kC j C k )g\circ f:=(c_{a^{-1}(T)},\otimes_{b\circ a)(i)=k}C_i\simeq \otimes_{b(j)=k}\otimes_{a(i)=j} C_i\to \otimes_{b(j)=k}C_j^'\to C_k^{''})

for 1kl1\le k\le l.

A monoidal (,1)(\infty,1)-category is defined to be a cocartesian fibration p:C N(Δ) opp:C^\otimes\to N(\Delta)^{op} such that:

  • For all n0n\ge 0, the associated functors C [n] C {i,i+1} C^\otimes_{[n]}\to C^\otimes_{\{i,i+1\}} determine an equivalence of (,1)(\infty,1)-categories
    C [n] C {0,1} ××C {n1,n} (C [1] ) n.C^\otimes_{[n]}\to C^\otimes_{\{0,1\}}\times \dots\times C^\otimes_{\{n-1,n\}}\simeq (C^\otimes_{[1]})^n.

where C [n] :=p 1([n)C^\otimes_{[n]}:=p^{-1}([n) denotes the fiber of the forgetful functor p:[C 1,,C n][n]p:[C_1,\dots,C_n]\mapsto [n] over [n][n].

In particular for S:={C,D}S:=\{C,D\} a set with two distinct we obtain:

Definition

Definition

Definition (relative nerve)

Let II be a category, let f:IsSetf:I\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 JN f(I)\Delta^J\to N_f(I) consists of:

  1. a functor s:JIs: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 Jf(σ(j ))\tau(J^\prime):\Delta^{J}\to f(\sigma(j^\prime)).

  3. satisfying properties.

mapping simplex: Let mapping simplexϕ:A 0A 1A n\phi:A^0\leftarrow A^1\leftarrow \dots\leftarrow A^n: Let be a composable sequence of maps of simplicial sets. The ϕ:A 0A 1A n\phi:A^0\leftarrow A^1\leftarrow \dots\leftarrow A^nmapping simplex of ϕ\phi be a composable sequence of maps of simplicial sets. The is denoted by mapping simplex of ϕ\phiM(ϕ)M(\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 MEnd [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 February 5, 2013 at 22:32:00 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.