Dear Andre,
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?
the quickest reference I can find is the very incomplete page http://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category but more importantly, the thesis referenced there. As I remark at the above page, Ross Street would know something about this. As to the other idea, someone (Toby Bartels perhaps) once said that explicitly equipping 'sets' with a notion of when elements are equal goes back to Cantor. I find thinking of the undergraduate (or high school) definition of equality of functions |R --> |R as a pale shadow of the idea of preset. Consider the collection of expressions denoting functions (even restricting to polynomials), this is a preset with the equality relation given by when one algebraic expression gives the same function as another algebraic expression. Best, David Roberts 2010/1/13 Joyal, André <joyal.andre@uqam.ca>:
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
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