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.

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    $\begingroup$ These questions are getting a bit repetitive. Do you have some overarching goal you'd like to share? $\endgroup$ – Kevin Carlson Feb 11 '16 at 0:15
  • $\begingroup$ I will add what i can to to my question here. $\endgroup$ – Ben Sprott Feb 11 '16 at 0:24
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    $\begingroup$ Would these questions be more suitable for mathoverflow? Also you are also going to have to restrict the number of objects you have in your category of categories. It is possible to have a category with a cardinality greater then $ | \mathbb{R} | $ and such categories are too large to have measures. $\endgroup$ – Q the Platypus Feb 11 '16 at 0:49
  • $\begingroup$ I guess I will take this to math overflow. $\endgroup$ – Ben Sprott Feb 11 '16 at 20:11

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