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The Stacks project

Lemma 101.39.5. Assume given a 2-commutative diagram

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[dd]_ j & \mathcal{X} \ar[d]^ f \\ & \mathcal{Y} \ar[d]^ g \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-z & \mathcal{Z} }

Choose a 2-arrow \gamma : z \circ j \to g \circ f \circ x. Let \mathcal{C} be the category of dotted arrows for the outer rectangle and \gamma . Let \mathcal{C}' be the category of dotted arrows for the square

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-{f \circ x} \ar[d]_ j & \mathcal{Y} \ar[d]^ g \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-z & \mathcal{Z} }

and \gamma . Then \mathcal{C} is equivalent to a category \mathcal{C}'' which has the following property: there is a functor \mathcal{C}'' \to \mathcal{C}' which turns \mathcal{C}'' into a category fibred in groupoids over \mathcal{C}' and whose fibre categories are categories of dotted arrows for certain squares of the form

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y & \mathcal{Y} }

and some choices of y \circ j \to f \circ x.

Proof. This lemma is a special case of Categories, Lemma 4.44.3. \square


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