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Question.

What, in your opinion, should replace the "?" in:

If (category)$=$(0-cell in CAT), then (bicategory)$=$(0-cell in ?).

Remarks.

  • "bicategory" as usual means "weak 2-category".
  • I am well aware that this is sort-of-opinion-based and does not have an uncontroversial precise answer, and is not asking much more than for mathematicians more experienced in category theory to suggest
    • what the category of all bicategories should mean precisely,
    • how the category of all bicategories should be denoted.
  • Still, I think there is some justification for asking this. It is useful in some contexts to be able to think of "category" as "0-cell in the category of categories", and there is use, too, for being able to do the same for "bicategory" instead of "category".

  • There are some issues what definition of "functor between two bicategories" to adopt, in the first place, but I will not get into this here (see the next remark). It can be argued that the definition should at least
    • make $\mathsf{CategoryOfBicategories}$ have a all finite products.
  • I do not give explanations of technical terms since I am curious what others think should be the "canonical" answer here.
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    $\begingroup$ What do you mean by a "weak bicategory"? Do you mean a bicategory in the sense of Bénabou? If so, bicategories and lax functors (called ''morphisms of bicategories'' by Bénabou) form a category, subcategories of which are obtained by restricting the morphisms to the pseudofunctors (or "homomorphims") or the strict functors. All three of these categories have finite products (and so have terminal objects). Furthermore, all three of these categories can be extended to 2-categories, whose 2-cells are "icons" defined by Lack, and they have finite products in the strict 2-categorical sense. $\endgroup$ – Alexander Campbell Jul 15 '17 at 17:35
  • $\begingroup$ @AlexanderCampbell: yes, here, "weak bicategory"$=$"bicategory in the sense of Bénabou". In particular, left unitor, right unitor and associator are only required to be natural isomorphisms, not identities. I wrote "weak" for emphasis. I know it is somewhat of a pleonasm, since the "bi" is usually taken to be sufficient to indicate the weak sense. Thanks for drawing attention to this, though, I will modify the question a bit. $\endgroup$ – Peter Heinig Jul 15 '17 at 17:54
  • $\begingroup$ And thanks for pointing out that my having opted for the desideratum "all finite products" renders the explicit mentioning of "terminal 0-cell" redundant. This was an oversight, resulting from the eternal issue of whether to write "all binary products" or "all finite products", conditions which oftentimes, in a certain context, are equivalent. You made me aware of a nice subtlety though, namely that this choice is not indifferent, the consequences of this choice are different. $\endgroup$ – Peter Heinig Jul 15 '17 at 18:05
  • $\begingroup$ @AlexanderCampbell: I understand that the category of bicategories in the sense of R. Street's 1980 Cahiers-article is some sort of standard $\mathsf{CategoryOfBicategories}$. Do you, roughly, agree with this? The notation $\mathsf{Hom}$ that is sometimes used for it seems not the most felicitous; did you meet alternatives for $\mathsf{Hom}$, and which do you recommend? Would you find $\mathsf{BiCat}$ instead of $\mathsf{Hom}$ acceptable? $\endgroup$ – Peter Heinig Jul 16 '17 at 8:49
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    $\begingroup$ The category of bicategories and pseudofunctors (first recognised by Bénabou, and, as you say, called $\textbf{Hom}$ in some articles of Street) is often called $\textbf{Bicat}$ (and indeed, this is what I use). The categories of bicategories and strict/lax functors are then correspondingly denoted $\textbf{Bicat}_s$ and $\textbf{Bicat}_l$. I do agree that pseudofunctors are in some sense the "correct" notion of morphism of bicategories, but that is not to impugn the other two kinds of morphism, both of which are useful. $\endgroup$ – Alexander Campbell Jul 16 '17 at 13:38
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Since it seems too complicated for this question to first ask the commenter whether they would like to make the comment an answer, I'll write one here myself (thanks to Alexander Campbell for confirming that $\mathsf{Bicat}$ is a usual notation): arguably the most usual $\mathsf{CategoryOfBicategories}$ is the category which is denoted $\mathsf{Hom}$ in Street's 1980 article in Cahiers de topologie et géométrie différentielle catégoriques, but the most usual notation nowadays is $\mathsf{Bicat}$. Subscripted variants thereof exist, to distinguish some-sort-or-full-subcategories of $\mathsf{Bicat}$, like e.g. $\mathsf{Bicat}_s$ and $\mathsf{Bicat}_l$ mentioned in a comment above, or $\mathsf{Bicat}_2$ in the sense of R. Garner, N. Gurski: The low-dimensional structures formed by tricategories. Math. Proc. Camb. Phil. Soc. 146 (2009),pp. 551--589.

And, to give one application of the usual notation, and another usual notation (namely $\mathsf{2Cat}$ for strict 2-categories), I mention that there is an adjunction

$$ \array{ & & & & \xrightarrow[]{\quad\mathrm{strictification}\quad} \hspace{-10pt} & & & & \\ & & {\huge\mathsf{Bicat}} & & \perp & & & {\huge\mathsf{2Cat}} & \\ & & & &\xleftarrow[\quad\mathrm{inclusion}\quad]{} \hspace{-10pt} & & & &\qquad. \\ }$$

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