Definition 84.27.1. Let $a : Y \to X$ be a morphism of simplicial schemes. We say $a$ is *cartesian*, or that *$Y$ is cartesian over $X$*, if for every morphism $\varphi : [n] \to [m]$ of $\Delta $ the corresponding diagram

\[ \xymatrix{ Y_ m \ar[r]_ a \ar[d]_{Y(\varphi )} & X_ m \ar[d]^{X(\varphi )}\\ Y_ n \ar[r]^{a} & X_ n } \]

is a fibre square in the category of schemes.

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