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.h... 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