Is there a particularly simple example of geometric descent? I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition.  
I'm not counting the definition of the arrow category $\mathbf{Cont}$ over $\mathbf{Top}$ from Vistoli's notes as particularly simple or familiar for the following reasons:


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*The fibred category $\mathbf{Cont}$ is not something that geometers, at least at a naive level, are normally interested in.  Perhaps category theorists are, but it seems rather unfamiliar (and it's not one of those things that initially seems unfamiliar and then turns out to be something you've been using all your life, such as fibred categories in general).  

*The example of descent was gluing various maps $X_i\to U_i$ together to make a map $X\to U$, where $(U_i)$ is a cover of the space $U$.  This involves gluing the $X_i$ along the pullbacks of the intersections of the $U_i$ inside $U$ to make a space $X$, and so involves the cocycle condition etc.  But this sort of gluing does not correspond to our usual notion of gluing topological spaces.  Normally, when we glue spaces $X_i$ there is no space $U$ with an open cover $U_i$ that we are mapping into - we just choose transition functions mapping between certain open subsets $X_{ij}$ of the spaces $X_i$, and glue them together to form a space $X$.  It looks like descent (we use the cocycle condition, etc.) but unless I am missing something and there is a canonical choice of $U,(U_i)$ that turns 'normal' space gluing into 'descent-style' space gluing, it isn't descent at all.
Here are my reasons for thinking that a simple example might exist:


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*Gluing and cocycle conditions pop up all over the place in geometry.  We can glue topological spaces, we can glue schemes and we can glue sheaves on spaces.  In none of those situations is there some category of 'base' objects keeping track of what we are doing, though.  

*Fibred categories are ubiquitous in geometry - we have the category of locally ringed spaces fibred over $\mathbf{Top}$, for example and $\mathbf{Top}$ is fibred over $\mathbf{Set}$, and there are many other examples.  


I just can't see a way to combine the two.  
 A: The arrow category is an extremely important example of a fibred category. (It is called the fundamental fibration for a reason!) In the case of $\mathbf{Top}$, the arrow category contains just about every variation you can think of the fibred category of fibre bundles: so, for instance, the arrow category contains as a fibred subcategory the fibred category of  vector bundles that are locally trivial and of some fixed rank.
Moreover, these inclusions are usually well-behaved with respect to descent along open covers, in the sense that e.g. the underlying topological space of realisation of a descent datum for the fibred category of vector bundles is the same as the realisation of the underlying descent datum for the arrow category. 
Finally, to see the connection with gluing objects, one notes that the realisation of descent data is often a special case of colimits for diagrams of shape $\mathbf{\Delta}^\mathrm{op}$, where $\mathbf{\Delta}$ is the simplex category. For instance, if you have an open cover $\{ U_i \}$ of $U$, we can form the simplicial topological space $B_{\bullet}$ where $B_n$ is the disjoint union of the $(n + 1)$-fold intersections, and then the canonical map $\varinjlim_n B_n \to U$ will be a homeomorphism. A descent datum for the arrow category over $\{ U_i \}$ is the same as a simplicial topological space $E_{\bullet}$ equipped with a simplicial map $E_{\bullet} \to B_{\bullet}$ satisfying certain conditions, and we can indeed identify the realisation with $\varinjlim_n E_n \to \varinjlim_n B_n$. On the other hand, gluing spaces together is also a special case of colimits of diagrams $\mathbf{\Delta}^\mathrm{op} \to \mathbf{Top}$: see this answer for some hints.
