Monads and the Barr-Beck Theorem (Rev #3)

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).

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 ϕ:A 0A 1A 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 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.


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 08:29:28 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.