Higher Grothendieck fibrations and the globular comma complex

Higher Grothendieck fibrations and the globular comma complex

[Harry Gindi]

Hello, I was thinking about the definition of a Grothendieck fibration
using adjunctions with respect to the comma category, and I seem to
have come upon an interesting construction.  I was wondering if it has
appeared in the literature:

Suppose f:B->A<-C:g is a diagram of strict ω-categories, let D^n
denote the n-globe, let ∂D^n be the boundary of D^n (which is the
empty strict ω-category when n=0).  Then for each n in N, let

f ↓_n g be the limit of the diagram

                         B^{D^n} x_{A^{∂D^n}} C^{D^n}
                                               ↓
A^{D^{n+1} → A^{D^n} x_{A^{∂D^n}} A^{D^n}


This construction is functorial in f, g, and n, and in particular,
functoriality in n makes

f ↓_{ - } g a globular strict ω-category.

Now suppose p: E → B is an isofibration of strict ω-categories  (that
is, it is a fibration in the folk model structure).

Then by functoriality, for each n in N we obtain maps

k_n: E^{D^{n+1}} → p ↓_n B compatible with the source and target
projections of the globular object, that is to say, the family of maps
is a morphism k of globular objects in strict ω-categories

Then I was wondering if anyone has tried to define an ω-cocartesian
fibration to be an isofibration of strict ω-categories such that  the
morphism of globular objects, k, has a left-adjoint right-inverse.

It seems like a natural construction/definition, so I'm sure someone
else has thought about it.  It also seems to generalize somewhat
straightforwardly to weak models of higher categories, provided they
can be cotensored with globes and we have a reasonable notion of
adjunction and pullback.

In the case n=0 (the indexing is weird because I wanted to describe it
as a globular object), we recover the usual notion of a cocartesian
fibration, and after that, I haven't worked out if it recovers
Hermida's definition of a cocartesian 2-fibration, but it seems like
it should.

Please let me know if there's any information in the literature about
this construction and its relationship with higher fibrations.

Thank you for your time and attention!

Your humble servant,

Harry J. Gindi


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