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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]