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Every groupoid is equivalent to a disjoint union of groups. In fact the inclusion of the sub-2-category of disjoint unions of groups into all groupoids is an equivalence. Hence the representation theo …

I what to know if there is a good generalization of `separated' for algebraic stacks? My usual stack reference, Anton Gerashchenko's stack notes, doesn't seem to provide an answer. … The main obstacle that I can see in defining separated for stacks is that the property of a map of schemes $X \to Y$ being separated does not appear to be local in the target. …

Since stacks form a 2-category, the natural thing is a "weak" or "homotopy" colimit. See the n-lab pages. … The inclusion from spaces into stacks doesn't preserve colimits, so even though this is a diagram of ordinary spaces, the colimit is an interesting stack. …

Yes, there is something to this effect. In fact there is a very general context for this. Since I know you are amenable to $\infty$-categories, I will use that language. The homotopy theory of spa …

I recommend Anton's course notes on Stacks as taught by Martin Olsson. … In a gerbe you are not patching together classifying spaces, you are patching together classifying stacks. Despite the common notation, there is a difference. …