more dagger problems

[John Baez <baez@math.ucr.edu>]

Juergen Koslowski wrote:

>This bicategorical view also clarifies that the star operation is an
>involution on 1-cells, while dagger is an involution on 2-cells.
>While the name ``{1,2}-involutive bicategory'' may be adequate,
>``{1,2}-involutive monoidal category'' is quite a mouthful.

I call them "monoidal categories with duals".  If you only have
your "star", I often call them "monoidal categories with duals
for objects".  If you only have your "dagger", I often call them
"monoidal categories with duals for morphisms".

They're a special case of a fascinating notion, "k-tuply monoidal
n-categories with duals", which so far only been precisely defined
for low values of n and k.  The "tangle hypothesis" proposes a nice
topological description of the free k-tuply monoidal n-category with
duals on one object.  Here are some places to read about this stuff:

John Baez and James Dolan, Higher-dimensional algebra and topological
quantum field theory, http://arxiv.org/abs/q-alg/9503002

John Baez and Laurel Langford, Higher-dimensional algebra IV: 2-Tangles,
http://arxiv.org/abs/math.QA/9811139

John Baez, Quantum computation and symmetric monoidal categories,
http://golem.ph.utexas.edu/category/2006/08/quantum_computation_and_symmet.html

One can also listen to lectures:

Eugenia Cheng, n-categories with duals and TQFT,
http://www.fields.utoronto.ca/audio/#crs-ncategories

The cases that have been precisely defined include:

n = 1, k = 0 categories with duals
n = 1, k = 1 monoidal categories with duals
n = 1, k = 2 braided monoidal categories with duals
n = 1, k = 3 symmetric monoidal categories with duals

n = 2, k = 0 weak 2-categories with duals
n = 2, k = 1 semistrict monoidal 2-categories with duals
n = 2, k = 2 semistrict braided monoidal 2-categories with duals

Here "weak 2-categories" means "bicategories" and "semistrict monoidal
2-categories" means "one-object Gray-categories".

For n = 1 we have up to 2 layers of duality (your "stars" and "daggers"),
while for n = 2 we have up to 3.

Best,
jb