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The Stacks project

Example 14.5.5. Suppose that X\to Y is a morphism of C such that all the pushouts Y\amalg _ X Y \amalg _ X \ldots \amalg _ X Y exist. Then we set U_ n equal to the (n + 1)-fold pushout, and we let \varphi : [n] \to [m] correspond to the map

(y \text{ in }i\text{th component}) \mapsto (y \text{ in }\varphi (i)\text{th component})

on “coordinates”. In other words, the map U_1 = Y \amalg _ X Y \to U_0 = Y is the identity on each component. The two maps U_0 = Y \to U_1 = Y \amalg _ X Y are the two coprojections.


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