Subject: categories: symmetry vs. duality

From: "Al Vilcius" <[email-masked]>
Date: Mon, 11 Jun 2001 12:50:02 -0400

Both symmetry and duality are important concepts, for sure;
both conjure up images of "2-ness" (or 2-folded-ness or 2-sided-ness ...).
However, their similarities and differences are problematic to me,
and I'm seeking either agreement or objection from this forum
to my remarks, plus a question on definition.

An over-simplified picture (perhaps trivial) that begins to distinguish
between these two fundamental aspects of 2-ness is the following:
symmetry:
.   ----------->
o                      o
.   ----------->        ie. confluence
vs.
duality
.   ---------->
o                      o
.   <-----------         ie. unity of opposites

"Symmetry is a vast subject, significant in art and nature"
said Hermann Weyl in his 1952 Princeton UP book "Symmetry".
As is well known, his mathematical approach culminates in the understanding
of symmetric configurations of elements as invariants under subgroups
of the group of automorphisms of their ambient structures.
This, of course, relates strongly to the famous Erlanger Program (1872)
of Felix Klein which sees geometry as the study of the properties of a space
that are preserved (invariant) under a given group of transformations,
whereby a geometry is distinguished by the group of transformations
under which its theorems remain true.

Duality is also a very important and far-reaching subject -
and certainly no need to recite further in this forum.

In terms of comparisons of similarities and differences,
I am tempted to attach the follow descriptions:
symmetry .......... vs. ............. duality
........2-ness................................2-ness
.. metric (or measure) related........order related
... visual................................complementary
..... causal parallel......................dialectic
.... syntactic..............................syntactic
.... concrete..................................abstract
.. invariance under transformation.....adjointness
.. geometric in character...........algebraic in character
...  confluence..........................unity of opposites

.... are there others that come to mind?

In the vernacular, symmetry and duality are sometimes used interchangeably,
both referring to 2-ness:  the quality, character or condition of being
two or twofold - a dichotomy.
Indeed it has been suggested that visual symmetry,
in the sense of exact correspondence of form and constituent
configuration on opposite sides of a dividing line or plane or about a
center or an axis, is a source of beauty as a result of this balance or
harmonious arrangement.
Alternatively, symmetry has also been seen more abstractly (vs. visually) as
a relationship of equivalence, identity, or characteristic correspondence
among constituents of an entity or between different entities, such as in:
the narrative symmetry of a novel. This level of abstraction is certainly
evident in the philosophical "Mind Body Problem" whose classical exposition
in the form of dualistic interactionism is due to René Descartes.

Both symmetry and duality should be seen as sources
of beauty in mathematics also,
however, in order to be addressed mathematically.
these should be precise.
The categorical notion of duality is captured precisely by adjointness,
or the unity and identity of opposites.
So finally, here is the question:
is there a (universal) categorical description of symmetry?

My hope is that precise formulations will allow me to better understand
heuristic notions such as wave-particle duality or symmetry-breaking etc.
in terms of (model) structures.

Thank you for your kind attention.

Al Vilcius
personal: [email-masked]
business: [email-masked]