I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction:
Throughout this chapter $\mathsf C$ denotes a fixed category with pullbacks.
Do they mean their entire article, i.e Chapter VIII?
Now the real question. Theorem 3.7 characterizes effective descent morphisms for particular types of categories and this has caused me some confusion.
An effective epimorphism in a 1-category is a regular epi which has a kernel pair (equivalently, an arrow with a kernel pair which is also its coequalizer). So an effective epi $f$ is one which gives rise to the obvious exact fork. To say this fork is exact is the same as saying it is a colimit diagram of for the kernel pair, which is in turn the 1-truncation of the Čech nerve of $f$.
One way to define descent data in general is as the homotopy limit of the cosimplicial image of its Čech nerve along a fibration $F$. If we're fibered in $n$-categories we may truncate above level $n$ before calculating the homotopy limit. This generalizes the sheaf and stack conditions.
Now if I understand Qiaochu Yuan's answer to this MO question, then for $F$ a 1-presheaf taking coequalizers to equalizers, TFAE:
- $f$ is an effective epi.
- $f$ is an effective descent morphism.
Theorem 3.7, however, seems to involve no fibration. How to translate it into the language above?