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*
= CX ), 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 evX :
<X*, X> → C,
composition with which gives bijections
A → X* <==>
<A, X> → C via
[f: A → X*] ···>
[<A, X> → C] =
evX · <f, idX>:
<A, X> → <X*, X> → C
or, even better, bijections between multi-maps
<A1, ..., An>
→ X* and multimaps
<A1, ..., An, X>
→ C
by the same procedure: f ···>
evX · <f, idX>.
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 iX: X → X**
corresponding by adjointness to idX*,
or, by dualization and symmetry, to
<X, X*> “→”
<X*, X> → C
(first “map”: symmetry;
last map: evX).
Even, duals
<X1, ..., Xk>*
for multi-objects
<X1, ..., Xk>
(k ≥ 0)
(C = <>*).