I've been recently looking into extending the Grothendieck construction to the case of a strict $\omega$-functor $F$ from a strict $\omega$-category $X$ to $\omega-\operatorname{Cat}$. By some recent developments (a paper of Dimitri Ara and Georges Maltsiniotis on defining the lax/oplax join and slice for strict $\omega$-categories), it has become pretty much a breeze to define the Grothendieck construction for such an $\omega$-functor as the oplax colimit of $F$ in $\omega-\operatorname{Cat}$, as it is simply defined to be the initial object of the oplax slice of $\omega-\operatorname{Cat}$ under $F$.

However, in a private conversation, I was informed of Scott Dyer's thesis, where he argues that in dimensions higher than $2$, things go rather bad and we end up in a situation without enough Cartesian lifts.

Here's the trouble: Dyer's thesis is pretty abstruse, and he uses a lot of nonstandard notation. It's also quite long. I'm wondering if anyone here might be familiar enough with it to save me some time reading through it only to find out that I'm barking up totally the wrong tree:

1.) Is Dyer's Grothendieck integral exactly the same construction as the aforementioned oplax colimit?

2.) Assuming the first question is true, has anyone come up with a loosening of either the definition of cartesianness for higher cells or the definition (as far as one exists) of a strict n-fibration. If fibrations in dimension n>2 are not the correct characterization of the image of the Grothendieck construction, has anyone come up with an alternative one since then?

Edit: Here's the paper

Dissertations, Theses, and Student Research Papers in Mathematics.67. digitalcommons.unl.edu/mathstudent/67 $\endgroup$3more comments