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

  Arens’s plot: dualize into a good dualizing object C (X ···> X* = C^X ), and then produce monoidal structure for the second-dual functor X ···> X**.
  Where? In a monoidal (or just a multi-linear) category where dualizing into C makes sense (Arens’s definitions and notations seem intentionally ambiguous).
  Needed along with X*, a given bilinear evaluation map ev_X : <X*, X> → C,
composition with which gives bijections A → X* <==> <A, X> → C via
[f: A → X*] ···> [<A, X> → C] = ev_X · <f, id_X>: <A, X> → <X*, X> → C
or, even better, bijections between multi-maps <A₁, ..., A_n> → X* and multimaps <A₁, ..., A_n, X> → C by the same procedure: f ···> ev_X · <f, id_X>.
  If dualizing (*) is self-adjoint, as it is in the sort of symmetric monoidal category that Arens seems to be in, there’s also a canonical map i_X: X → X** corresponding by adjointness to id_X*, or, by dualization and symmetry, to
<X, X*> “→” <X*, X> → C (first “map”: symmetry; last map: ev_X).
(Using: bil(<A, B>, C) ~ hom(A ⊗ B, C) ~ hom(B ⊗ A, C) ~ bil(<A, B>, C).)