Re: additions

[selinger@mathstat.dal.ca (Peter Selinger)]

In defense of Andre's list, the explanation he gave for his original
list was that subjects in category theory become hot from time to time
in response to factors such as new developments outside category
theory. The list was supposed to be a list of categorical subjects,
not a list of the respective developments that have inspired their use
and advancement.

The current use of category theory in quantum foundations is clearly
an interesting development, and has inspired new work in category
theory. But I would still be comfortable, for the time being, in
classifying this new work as falling within the existing subjects of
"monoidal categories" and "category theory and computer science" on
Andre's growing list. Recently also "topos theory" due to the work of
Andreas Doering, Klaas Landsman, and others on topos models for basic
physics.

As for the completeness result that Thorsten mentioned, the reference
is:

 M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector
 spaces are complete for traced symmetric monoidal categories".  In
 Pillars of Computer Science: Essays Dedicated to Boris (Boaz)
 Trakhtenbrot on the Occasion of His 85th Birthday, Springer LNCS
 4800, pages 367-385, February 2008.

The result is that an equation holds in all traced symmetric monoidal
categories if and only if it holds in finite dimensional vector
spaces. An immediate corollary is that the analogous result holds for
compact closed categories.  A simplified proof, and extension to
dagger compact closed categories (w.r.t. finite dimensional Hilbert
spaces), can be found here:

 P. Selinger, "Finite dimensional Hilbert spaces are complete for
 dagger compact closed categories". Extended abstract, to appear in
 Proceedings of the 5th International Workshop on Quantum Physics and
 Logic (QPL 2008), Reykjavik, 2010.

Merry Christmas to all, -- Peter


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]