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When we deal with usual category $\mathbf A$ we require that Ob$(\mathbf A)$ (i.e., collection of objects of category $\mathbf A$) and Mor$(\mathbf A)$ (i.e., collection of morphisms between objects in category $\mathbf A$) form a class (proper or no).

Then if we deal with metacategory $\mathbf A$ we say that Ob$(\mathbf A)$ is a conglomerate and Mor$(\mathbf A)$ so is.

To form a conglomerate we require:

  1. every class is a conglomerate,
  2. for every “property” P, one can form the conglomerate of all classes with property P,
  3. conglomerates are closed under analogues of the usual set-theoretic constructions, and
  4. the Axiom of Choice for Conglomerates; namely for each surjection between conglomerates $f:X\rightarrow Y$, there is an injection $g:Y\rightarrow X$ with $f\circ g=id_Y$.

So the question is: can we continue this hierarchy further to form the metacategory of all metacategories and so on?

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    $\begingroup$ This use of the term "quasicategory" is horribly archaic; please avoid it. $\endgroup$ – Mike Shulman Oct 5 '17 at 15:48
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    $\begingroup$ Mike is too polite to self-advertise, but he wrote a great exposition on (set-theoretical) foundational issues in category theory: arxiv.org/abs/0810.1279 $\endgroup$ – Myself Oct 5 '17 at 16:05
  • $\begingroup$ @Myself, i’ve never seen this paper, so i’ll check it! $\endgroup$ – A. Gonus Oct 5 '17 at 19:11
  • $\begingroup$ You should also be careful with the adjective "regular" here. Perhaps "usual" would be more appropriate. It's not a big deal, but I was a bit puzzled at first. $\endgroup$ – Arnaud D. Oct 6 '17 at 12:40
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    $\begingroup$ @PeterHeinig, yes, I’ve talked about the operations which I mentioned in my question, so you are right. $\endgroup$ – A. Gonus Oct 7 '17 at 9:16

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