Why there is no 3-category or tricategory of bicategories?

I recently asked this question on MSE.

So I want to move it here in hope to gain a more wordy answer.

I have read around about bicategories, lax functor, lax natural transformation and modifications. I know that we have a 1category of Bicategories and lax functors. I know why we do not have a bicategory or 2category of bicategories, lax functor, and lax natural transformations, and I know that using ICONS instead of lax natural trasformations solves this problem. (or oplax, it doesn’t matter)

What I cannot see is why we do not get a 3categories or tricategories of bicategories, Lax functors, ICONS and modifications? What fails? Where can I find a reference about it?

MathOverflow Asked by Lolman on December 27, 2020

1 Answers

One Answer

Edit: In this answer I missed that the question was about lax functors rather than pseudofunctors. See comments below.

Such a tricategory does exist, and in fact it is part of a richer structure. In Garner-Gurski The low-dimensional structures formed by tricategories (arxiv), Corollary 12 constructs a locally cubical bicategory of bicategories. This is a bicategory enriched over the monoidal 2-category of pseudo double categories, containing the following structure:

  • its 0-cells are bicategories
  • its 1-cells are pseudofunctors
  • its "vertical 2-cells" are icons
  • its "horizontal 2-cells" are pseudonatural transformations
  • its 3-cells are "cubical modifications".

Thus, if we discard the vertical 2-cells we obtain the usual tricategory of bicategories (although one has to do a bit of work to "lift" the coherences, which start life as icons, to pseudonatural transformations). If we instead discard the horizontal 2-cells, I believe we obtain the tricategory you're after.

It is a particularly strict sort of tricategory. This construction exhibits it as a bicategory enriched over the monoidal 2-category of strict 2-categories, but it might in fact be a strict 3-category; I have not checked carefully.

I suspect that your next question might be why no one has pointed this out before. Probably the answer is that no one had a use for it. One of the purposes of introducing icons was to reduce the categorical dimension of the structure containing bicategories, so putting the modifications back in would defeat that purpose. Also, modifications between icons may seem a priori to be of limited interest, since their components are endomorphisms of identity 1-cells — although of course such a judgment always evaporates when someone finds an application of them! Finally, the locally cubical bicategory seems more useful for most purposes: its categorical coherence dimension is equally limited (being an enriched bicategory, rather than a tricategory), while it contains strictly more information.

Answered by Mike Shulman on December 27, 2020

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