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I was playing a bit the 2-category Cat trying to have a better understanding of the notion of a 2-category (strict I guess). The usual definition of a category that I use assumes that $Hom(A,B)$ is a set.

What is an analogue of that condition in 2-categories? I guess you need to have some size restrictions in order to have a higher-Yoneda. I think the class of natural transformations between two functors is not a set in general, am I right?

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    $\begingroup$ The obvious analogue is to have the hom-categories be small. $\endgroup$ – Zhen Lin Mar 5 '16 at 14:22
  • $\begingroup$ But.. then Cat is not a 2-category in that sense because $Fun(C,D)$ is not necessarily small. What is Cat then? $\endgroup$ – Abellan Mar 5 '16 at 14:54
  • $\begingroup$ I am personally fine with having the hom categories be locally small. Asking them to be small, or even essentially small, rules out too many interesting examples. $\endgroup$ – Qiaochu Yuan Mar 5 '16 at 18:09
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    $\begingroup$ For me, $\mathbf{Cat}$ is the category of small categories. $\endgroup$ – Zhen Lin Mar 5 '16 at 21:27
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Paul Blain Levy has a short (1-page) note on the topic.

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In this post, Cat refers to the 1-category of small categories.

I believe what you are looking for can be naturally described in the language of enriched category theory:

  • the "usual definition of category" you use translates to "Set-enriched category",
  • the corresponding notion of strict 2-category translates to "Cat-enriched category".

And as usual, you can adjust to the relevant notion of universe needed. For example, rather than enrich in the (large) category of small categories, you could enrich in:

  • the (superlarge) category of large categories,
  • the (superlarge) category of large, locally small categories.

The latter might introduce some awkwardness since it's not a closed category.

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