Miles Gould’s work looks interesting. I guess it must be related to this work of Steve Lack (which I think is written up here)?

> Thinking about what John wrote, I realise that I made a mistake. I was
> wrong to claim that Bicat is a Gray-category:
> the problem is that post-whiskering with a pseudo-functor is not
> 2-functorial, even when the objects are 2-categories. (It looks as though
> Tom’s section III.3 contains the same error? Note that the 1-cells of his
> 2-Cat are pseudo-functors rather than 2-functors.)

Guilty (of the error, as well as the appalling notation. It was a long time ago.)

There’s a less subtle way in which Bicat fails to be a Gray-category. This was one of the points I was trying to make in that error-riddled essay. Here I’m assuming that your 2-cells in Bicat are *weak* transformations. In that case, vertical composition of them is not associative or unital. However, it is if you restrict to the full sub-thing of Bicat whose 0-cells are the strict 2-categories.

Miles Gould has been working on the st construction. He’s sticking to low dimensions for now, but has shown that it works for a range of categorified algebraic theories, not just monoidal categories. He also observes that (if you work in the unbiased setting) st is simply the left adjoint to the inclusion of strict structures into weak structures. See his notes. A polished version should appear on the arXiv before too long.