William James <wjames@arts.adelaide.edu.au> writes:
I suppose the answer is that monicity is relative to a category, but what supports this as a claim?
It seems to me that category theory takes the sensible viewpoint that mathematical entities (e.g. objects and morphisms) only become interesting through their relationship with other entities. Every arrow looks just like every other arrow if we consider it in isolation. Every arrow in a category C is an image of the what James Dolan calls the "walking arrow" --- the nonidentity morphism in the free category C0 on a single morphism --- under some functor F: C0 -> C. Studying an arrow in isolation is just like studying the walking arrow, which is completely dull. The fun begins only when we have a bunch of arrows and start composing them. This is one reason why I think n-category theory should be useful in physics problems like quantum gravity, where it only makes sense to speak of where or when an event occurs relative to other events, not with respect to some spacetime manifold of fixed geometry. For some of the technical apsects of how this might go, see: John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105. Louis Crane, Clock and category: is quantum gravity algebraic?, J. Math. Phys. 36 (1995), 6180-6195. These both appeared in a special issue on diffeomorphism-invariant physics.