categories fibered in small categories

categories fibered in small categories

[Andrew Salch]

I have a question for the categories list: it is well-known (and proven
e.g. in Hollander's PhD thesis) that categories fibered in groupoids over
a small category C are equivalent to lax presheaves of groupoids on C. I
would like to use the generalization of this result in which the word
"groupoids" is replaced throughout by "small categories." It is not hard
to write out how this proof goes, but I suspect there is some vast
generalization of this, e.g. with the word "groupoids" replaced by
"quasicategories" or something of that nature, which somebody has already
proven, and in that case I would prefer to cite the more general result.
Does anyone know if something like this is already in the literature?

Thanks,
Andrew S.


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

Re: categories fibered in small categories

[Urs Schreiber]

On Sun, Jun 7, 2009 at 1:37 AM, Andrew Salch<asalch@math.jhu.edu> wrote:

> it is well-known (and proven
> e.g. in Hollander's PhD thesis) that categories
> fibered in groupoids over a small category C
> are equivalent to lax presheaves of groupoids on C.
[...]
>I suspect there is some vast generalization of
> this, e.g. with the word "groupoids" replaced by
> "quasicategories" or something of that nature,
> which somebody has already proven, and in that
> case I would prefer to cite the more general result.
> Does anyone know if something like this is already
> in the literature?

Yes, see section 3.3.2 of Jacob Lurie's "Higher Topos Theory" for the
statement for "quasicategory valued presheaves".


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