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The Stacks project

Definition 4.35.6. Let \mathcal{C} be a category. The 2-category of categories fibred in groupoids over \mathcal{C} is the sub 2-category of the 2-category of fibred categories over \mathcal{C} (see Definition 4.33.9) defined as follows:

  1. Its objects will be categories p : \mathcal{S} \to \mathcal{C} fibred in groupoids.

  2. Its 1-morphisms (\mathcal{S}, p) \to (\mathcal{S}', p') will be functors G : \mathcal{S} \to \mathcal{S}' such that p' \circ G = p (since every morphism is strongly cartesian G automatically preserves them).

  3. Its 2-morphisms t : G \to H for G, H : (\mathcal{S}, p) \to (\mathcal{S}', p') will be morphisms of functors such that p'(t_ x) = \text{id}_{p(x)} for all x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}).


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