Definition 85.27.1 (0249)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.27: Descent in terms of simplicial schemes
Definition 85.27.1 (cite)

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Definition 85.27.1. 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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