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**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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