How do various notions of natural transformation relate to various notions of homotopy in $2Cat$?

Question

In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the answer to my question is rather standard for people working daily with $2Cat$ as a $2$- or $3$-category, and I think I may know some people who could answer my question if I ask them privately. However, it seems to me it could be useful to have an answer to that question on MO for the record (and so as to allow those knowledgeable people to earn much sought-after MO reputation and badges).

It is standard to view natural transformations between functors as categorical homotopies. There has been a question about this viewpoint on MO before. What I would like to know is the relationship between higher natural transformations and homotopies.

The starting point is that, given two categories $A$ and $B$ and two functors $F$ and $G$ from $A$ to $B$, a natural transformation from $F$ to $G$ is the same thing as a functor $\Delta_{1} \times A \to B$ which makes the obvious diagram commutative, where $\Delta_{1}$ is the category associated to the naturally ordered set $\{0,1\}$.

Now, if we replace $Cat$ by $2Cat$, we have a whole bunch of possible variants. For instance, if $\mathcal{A}$ and $\mathcal{B}$ are two $2$-categories and $F$ and $G$ are two $2$-functors from $\mathcal{A}$ to $\mathcal{B}$, then a lax natural transformation from $F$ to $G$ gives a lax $2$-functor $\Delta_{1} \times \mathcal{A} \to \mathcal{B}$ which makes the obvious diagram commutative. But we could ask whether a lax transformation between lax functors gives such a lax functor too, and whether these two notions are equivalent in this setting, as in the case of $Cat$ (I have just stated one implication only, and just for strict $2$-functors). The question which arise could therefore phrased as:

What are the objects in $2Cat$ analogous to $\Delta_{1} \times A$ in the case of $Cat$, with respect to strict transformations between strict $2$-functors, lax transformations between strict $2$-functors, lax transformations between lax $2$-functors?

I guess this is related to the Gray tensor product, but Gray's style is, woe is me, undecipherable to me. The related $nLab$ page seems to help but not to answer exactly this question (I may well be mistaken).

A related question is the one relating the $2$-categorical viewpoint and the $1$-categorical viewpoint:

What kind of transformations in $2Cat$ gives what kind of homotopies in $2Cat$? (Here, I primarily think to "homotopy" as an arrow from the product of $\Delta_{1}$ with something, but I do not want to restrict to this case if it turns out this is not the right generalization.)

This last question does not ask for the universal objects, but rather how various maps (the "various" refering to the degree of laxness) with various universal objects as domain relate to various (same remark) notions of transformations. I have routinely worked with $2Cat$ from the $1$-categorical viewpoint, but much less from the $2$-categorical viewpoint (let alone the dreaded $3$-categorical viewpoint).

If some people feel comfortable with the general setting of $nCat$, I do not have any objection, although I would mind if it should obfuscate what happens in the special case of $2Cat$.

Many thanks in advance!

Answer 1

For strict transformations between strict 2-functors, you can just use the cartesian product $\Delta_1\times A$.

For lax transformations between strict 2-functors, this is what the lax version of the Gray tensor product does. The nLab page is mostly about the pseudo version (which corresponds to pseudo natural transformations), but if you replace "pseudo" with "lax" then the defining isosmorphism $$2Cat(B\otimes_l C, D) \cong 2Cat(B, Lax(C,D))$$ gives you what you want if you take $B=\Delta_1$, where $Lax(C,D)$ denotes the 2-category of strict 2-functors and lax natural transformations.

For lax transformations between lax 2-functors, we can invoke the fact that for any 2-category $C$, there is a 2-category $Q_l C$ such that strict 2-functors $Q_l C \to D$ are in bijection with lax 2-functors $C \to D$. Then $\Delta_1\otimes_l Q_l C$ will have the property that strict 2-functors out of it are in bijection with lax transformations between pairs of lax 2-functors out of $C$. I doubt there is a way to get this to work if you want to look at lax functors out of the cylinder object, though.