Commutativity considerations
For
A a monoid in symmetric monoidal category, write • (not μ)
for its bilinear multiplication map
A ×
A →
A (Arens’s
suggestive/ambiguous product notation).
Using
a and
b as variables from
A,
φ as variable from
A*, and
S and
T
as variables from
A**, we have the following stages
towards the Arens multiplication on
A**:
|
•: A × A → A |
(a, b) ··> a • b |
a • b = •(a, b) |
|
•†:
A* × A → A* |
(φ, a) ··> [b ··> φ(a • b)
= (φ •† a)(b)]
|
φ •† a = φ(a • –) |
|
•††:
A** × A* → A* |
(T, φ) ··> [a ··> T(φ •† a)
= (T •†† φ)(a)] |
T •†† φ
= T(φ •† –) |
|
•‡
= •†††:
A** × A** → A** |
(S, T) ··> [φ ··> S(T •†† φ)
= (S •††† T)(φ)] |
S •‡ T
= S(T •†† –) |
Thus,
(ev
a •
‡ T)(φ) = ev
a(
T
•
†† φ)
= (
T •
†† φ)(
a)
=
T(φ •
† a)
=
T(φ(
a • –)), and
(S •‡ evb)(φ)
= S(evb •†† φ)
= S(φ(– • b)). (Indeed,
ev
b •
†† φ
= φ(– •
b) — proof:
∀a :
(evb •†† φ)(a)
= evb(φ •† a)
= (φ •† a)(b)
= φ(a • b).)
Hence (dropping daggers) ev
a •
T =
T • ev
a for
a in the center of
A (that’s all
a, for
A commutative), since
∀φ:
(eva • T)(φ) = T(φ(a • –))
= T(φ(– • a)) = (T • eva)(φ).
But, even for
A commutative,
S •
T =
T •
S
doesn’t follow for other
S.
Example: ...
F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus