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Stream: learning: questions

Topic: Graded bicategory

fosco (Nov 09 2022 at 22:11):

I have encountered the notion of "graded bicategory" defined as follows: image.png

I don't know exactly what a bifoncteur discrètement 2-cofibrant could possibly mean, in the sense that I don't know exactly what the author (R. Guitart, from "Tenseurs et machines": it can easily be found online, but the general topic of the paper is not related to this question at all...) means with that "2-"... Certainly it is some sort of discrete opfibration over $\cal V$ regarded as a bicategory with a single object; if this is true, it consists of a family of functors ${\cal B}(X,Y)\to \cal V$. But now, what is the discreteness (of the fibers?) in question?

(For those who do not now French, here's a literal translation: "a bifunctor (=functor of bicategories) $B : {\cal B}\to{\cal V}$ which is discretely 2-cofibrant is called a graded $\cal V$-bicategory": not much context is given, the notion is used -from page 46 onwards-, but not explained.)

fosco (Nov 09 2022 at 22:11):

(PPS : @Beppe Metere might be of some help curb my dumbness!)

fosco (Nov 10 2022 at 10:29):

Ah, I think I know what's going on here