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I am reading the 2008 book "Institution Independent Model Theory" by Razvan Diaconescu and looking up what the category ℂat was it states in the symbol Index

ℂat: the hyper-category of categories as objects and functors as arrows

But it nowhere defines what a hyper-category is. Is this just a way of saying that it is a Category containing Categories?

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It's analogous to the idea of a "hyper-class," which is to classes what classes are to sets. Basically, in an appropriate background theory, a hypercategory consists of a hyperclass of objects and a hyperclass of morphisms, as opposed to a class of objects and a class of morphisms (= category) or a set of objects and a set of morphisms (= small category).

Note that to even talk about such things, we need to enrich our background set theory in a manner similar to how we move from ZFC to NBG. However, we can do this in a "conservative way," so this isn't as big a deal as it sounds.

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  • $\begingroup$ A so essentially it's very big. :-) $\endgroup$ – Henry Story Sep 10 '18 at 20:13
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    $\begingroup$ @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc. $\endgroup$ – Noah Schweber Sep 10 '18 at 20:13
  • $\begingroup$ Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th $\endgroup$ – Henry Story Sep 11 '18 at 11:14

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