In the study of effective descent morphisms in a category 𝒞 with pullbacks, the standard technique is to embed 𝒞 in a larger (simpler) category 𝒟 with pullbacks where effective descent morphisms are well-understood, as this allows us to utilize one of many reflection theorems along the given embedding. Such results provide sufficient conditions for a morphism to be of effective descent in 𝒞, but in some cases a characterisation result is desirable.
To obtain such characterisation results, one must ask whether the embedding 𝒞 → 𝒟 preserves effective descent morphisms, a problem that has not received much attention. In this presentation, we will talk about novel methods that allow us to study whether a given functor 𝐹∶ 𝒜 → ℬ preserves effective descent morphisms, even in settings where the underlying categories may not have all pullbacks.
Our results are particularly well-suited in the setting where 𝐹 is a fibration of categories; indeed, our main contribution is that any fibration that preserves pullbacks also preserves effective descent morphisms. We also give some conditions for opfibrations and other families of functors to preserve effective descent morphisms, as we can - in certain cases - use our methods to turn reflection results into preservation results. In general, we provide a method that allows one to check whether suitable functors preserve effective descent morphisms.
This is joint work with F. Lucatelli Nunes.