C.M.S. • Fredericton, NB • June 2010
Meditations on Arens Multiplication
   F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus

  We fix a dualizing object C and write A* = C^A. The intent:
maps <X, Y, ..., Z> → A* = C^A should amount to maps <X, Y, ..., Z, A> → C.
  In particular, there is ev_A: <A*, A> → C corresponding to id_A*: A* → A* .
  From A_i , μ: <A₁, ..., A_n> → A₀, Arens’s μ^‡: <(A₁)**, ..., (A_n)**> → (A₀)**
is the last of n+1 2-step †-stages: the steps of †-stage 1 are: (1) the compositum
ev_A0·<id_A0*, μ>: <(A₀)*, A₁, ..., A_n–1, A_n> → <(A₀)*, A₀> → C (step 1) and
(2) the associated multinear μ^†: <(A₀)*, A₁, ..., A_n–1> → (A_n)* (step 2).
(Loosely: μ^†(φ, a₁, ..., a_n–1)(a_n) = ev_A0(<φ, μ(a₁, ..., a_n)>) = φ(μ(a₁, ..., a_n))).
  Iterate n ×: (μ^†)^† = μ^††: <(A_n)**, (A₀)*, A₁, ..., A_n–2> → (A_n–1)*, ...,
..., (μ^†n–1)^† = μ^†n: <(A₂)**, ..., (A_n)**, (A₀)*> → (A₁)*, and finally
(μ^†n)^† = μ^†n+1 =_def μ^‡: <(A₁)**, ..., (A_n)**> → (A₀)** (naturally in all A_i).
  NB: for μ: A → B, μ^† = μ*: B* → A* and μ^‡ = μ^†† = μ**: A** → B**; for
a: <> → A, ev_A·<id_A*, a>: <<>, A*> = <A*> = <A*, <>> → <A*, A> → C
and (w/ lgos) a^‡ = a^† = ev_a: <> → A** ( = i_A(a) in symmetric monoidal case).