In an attempt to recruit the help of a friend from school, he writes 
this in an email in response:


So, about your question, I've actually never heard
of a lattice-ordered abelian group, so I don't think I
can help you there. I can tell you about the
connection of category theory to physics, though
(although you may already know this): when you talk
about open string theory (i.e. adding D-branes to the
theory), depending on whether you consider the A or B
twist, the D branes are supposed to form a derived
Fukaya category for the A twist, or a category of
derived coherent sheaves on the B twist. In
categorical language, the objects are the D branes,
and the morphisms are (open) strings stretching
between D branes. If you wanted to then make some
(tenuous at best) connection to the real universe,
assuming that string theory is actually true, since
all particles are supposed to be strings (strings are
a subset of D branes), this means that theoretically
the entire universe could be described by a category
of D branes. The problem with this, though, is that D
branes are not fully described by even the derived
Fukaya/coherent sheaf setup, so before that kind of
connection can be made, (1) string theory has to be
proven true, (2) a complete mathematical description
of D branes has to be worked out.


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