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?

  • $\begingroup$ To ask the obvious question: does Steve Lack address it in his paper Icons, Appl Categor Struct 18, 289–307 (2010). doi.org/10.1007/s10485-008-9136-5? $\endgroup$ – David Roberts Oct 28 '20 at 1:02
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    $\begingroup$ @DavidRoberts I stubled upon that paper but there are just two instance of the word "modification" and are presented as the third level of a category with pseudonatural transformations. After presenting the ICONS he doesn't delve too much into higher levels. $\endgroup$ – Lolman Oct 28 '20 at 11:44
  • $\begingroup$ Apparently ICON is an acronym: "it is an Identity Component Oplax Natural-transformation". $\endgroup$ – Samantha Y Nov 6 '20 at 16:49
  • $\begingroup$ Right, but it's not usually capitalized. $\endgroup$ – Mike Shulman Nov 6 '20 at 18:16

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.

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    $\begingroup$ Thank you very much for the link to Gurski's paper. I may fail to see how from corollary 12 we can obtain lax functors from pseudo-functors by discarding the horizontal 2-cells. On the other hand the main result of the paper, stated in theorem 7 and "proved" in corollary 26, seems to containg the wanted result if I were to consider only the bicategories in that tricategory. Thus unless I made some kind of terrible blunder in understanding the paper, the stated tricategory really exists explicitly statd. Thank you. $\endgroup$ – Lolman Nov 10 '20 at 12:54
  • $\begingroup$ How embarrassing! I totally missed that you were asking specifically about lax functors rather than pseudo ones. You're right, the locally cubical bicategory of bicategories does not extend to one containing lax functors, for the same reason that there is no tricategory of lax functors and arbitrary pseudonatural transformations: you can't whisker a pseudonatural transformation (or a lax or colax one) by a lax functor. However, I'm glad you were able to find what you were looking for elsewhere in the Garner-Gurski paper. $\endgroup$ – Mike Shulman Nov 10 '20 at 16:49
  • $\begingroup$ Apparently I misread and got confused. After more carefully inspecting of the arcticle I found that this cannot answer the question because the "pseudo-Icon modifications" introduced here do not degenerate to normal modifications when considered between bicategories. At least I fail to see it explicitly. $\endgroup$ – Lolman Nov 13 '20 at 20:18
  • $\begingroup$ Ah yes, it looks like you are right. So your particular question is still open. $\endgroup$ – Mike Shulman Nov 14 '20 at 16:12

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