Consider a family of continuous maps $(U_i\to U)$. For a family of bundles $(X_i\to U_i)$, TFAE:
- The family is a pullback of a single bundle $X\to U$;
- There exist transition isomorphisms satisfying the cocycle condition.
Thus, for topological spaces, existence of descent data for a fixed object is equivalent to the object being in the essential image of the familial pullback $\mathsf C_{/U}\to \prod_i\mathsf C_{/U_i}$.
I am wondering whether and when the category of descent data is actually equivalent to the essential image of this familial pullback functor.
More generally, consider a pseudofunctor $F:\mathsf C^\text{op}\to \mathsf{Cat}$. Given a family of arrows $(U_i\to U)$, define the category $\mathsf{Desc}(F,(U_i\to U))$ of descent data as the 2-limit of the diagram below, where the arrows are familial pullback functors and $U_{ij},U_{ijk}$ are the obvious multi-pullbacks. $$\prod_iF(U_i)\rightrightarrows\prod_{ij}F(U_{ij})\substack{\longrightarrow\\[-1em] \longrightarrow \\[-1em] \longrightarrow} \prod_{ijk}F(U_{ijk})$$
When is there a canonical equivalence between the essential image of the familial pullback $F(U)\to \prod_iF(U_i)$ and the category of descent data for $F$ along $(U_i\to U)$ and how to prove it?
Remark. Note I am not asking when the category of descent data is equivalent to the domain of the familial base change. I am hoping for much much weaker conditions than the family $(U_i\to U)$ being of effective descent.
This may not be an answer, but definitively too long for a comment.
The way I understand descent theory is the following : if you have a cover $\{U_i\rightarrow U\}$ and objects $(X_i)$ on the $(U_i)$ then the question is not "Does the family $(X_i)$ comes from an $X$ on $U$", but rather "Do I have something on $(X_i)$ making it an object on $U$". And the answer may be
In other words, you have a functor $F(U)\rightarrow\prod F(U_i)$ which factors as $$ F(U)\rightarrow\operatorname{Desc}(F,\{U_i\rightarrow U\})\rightarrow\prod F(U_i)$$ and you start with a family $(X_i)\in\prod F(U_i)$. Then there are two different problems :
So you seem to be interested on the first point only. But I don't think the object satisfying this condition might organized in a interesting category (other that the full category of $(X_i)$ which have a descent data). In other words, if you consider only families $(X_i)$ which has isomorphisms $X_i|_{U_{ij}}\simeq X_j|_{U_{ij}}$ satisfying a condition, what morphisms do you want between to such families if you don't specify these isomorphisms (this extra-structure) ?
You ask about the essential image of $F(U)\rightarrow\prod F(U_i)$, but not necessarily the full essential image. I just don't know what it means. The image of a functor is not a category in general.
Let's have a look at an example where descent hold so we can safely replace $\operatorname{Desc}(F,\{U_i\rightarrow U\})$ by $F(U)$.
Galois descent of vector spaces. If $L/k$ is a (Galois) extension, the functor $F(k)\rightarrow F(L)$ is essentially surjective : every $L$ vector space has a $k$-structure. The whole point is the extra-structure. Now take two $\mathbb{C}$-vector $V,W$ spaces of dimension 2. What morphisms do you want to consider between them so that you can say : "they are morphisms between two $\mathbb{R}$-structure on them" ?
Do you want to consider all morphism ? Because after all, for all morphism $f:V\rightarrow W$ there are Galois semi-linear action on $V$ and $W$ making $f$ equivariant. In that case, your image category will be the full $F(L)$ but this is not equivalent to $F(k)=\operatorname{Desc}(F,L/k)$.
Or do you want to consider only equivariant morphism after choosing for every vector space a specific Galois semi-linear action ? That does not seem very canonical, and worse : the functor $F(k)=\operatorname{Desc}(F,L/k)\rightarrow F(L)$ does not factor through this "image" anymore.