Is there a suitable way to restrict the functors between objects (and I suppose the objects themselves) of a category of categories such that we have a cauchy complete category. Can we find a complete category this way?
The reason I am asking about this is that I would like to define a measure on a category of categories. By a measure, I mean a map from the category into the reals. I am not sure about this at all,but I think that if we have diagrams with (co)limits we could treat them like subsets and define a measure for them. The cauchy completeness is important for being able to treat the category like a measurable set.
Also, I want to define a metric on a cat of cats that defines how similar two categories are. For instance, a subcategory has some similarity to its parent and any category is exactly like itself.
We have found that subcategory is too restrictive to have Cauchy completeness, in that, a cat of cats and embeddings is not complete.