Dear All, I cannot imagine a category without an equality relation between the objects of this category. Ok, I may have been brainwashed by my training in mathematics at an early age. But more seriously, I think that the equality relation is inseparable from the idea of a set. I do not understand what a preset is: http://ncatlab.org/nlab/show/preset Two things are equal if they are the same, if they coincide (whatever that mean!). Without this notion, an element of a set has no identity, no individuality. Of course, a set is often constructed from other sets, as in arithmetic with congruence classes. I am fully aware that the equality relation between the objects of a category is not preserved by equivalences in general. But the art of category theory consists partly in knowing which construction on the objects and arrows of a category is invariant under equivalences. I would like to propose a test for verifying if the notion of category can be freed from the equality relation on its set of objects. The equality relation on an ordinary set S is defined by the diagonal S-->S times S. The objects of a symmetric monoidal category have no diagonal in general, ie no coalgebra structure. The test: Can we define a notion of category internal to a symmetric monoidal category without using a coalgebra structure on the object of objects? Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]