The Stacks project

Definition 88.10.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $. We define conditions (S1) and (S2) on $\mathcal{F}$ as follows:

  1. Every diagram in $\mathcal{F}$

    \[ \vcenter { \xymatrix{ & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & A_2 \ar[d] \\ A_1 \ar[r] & A } } \]

    in $\mathcal{C}_\Lambda $ with $A_2 \to A$ surjective can be completed to a commutative diagram

    \[ \vcenter { \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } } \]
  2. The condition of (S1) holds for diagrams in $\mathcal{F}$ lying over a diagram in $\mathcal{C}_\Lambda $ of the form

    \[ \xymatrix{ & k[\epsilon ] \ar[d] \\ A \ar[r] & k. } \]

    Moreover, if we have two commutative diagrams in $\mathcal{F}$

    \[ \vcenter { \xymatrix{ y \ar[r]_ c \ar[d]_ a & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[\epsilon ] \ar[r] \ar[d] & k[\epsilon ] \ar[d] \\ A \ar[r] & k } } \]

    then there exists a morphism $b : y \to y'$ in $\mathcal{F}(A \times _ k k[\epsilon ])$ such that $a = a' \circ b$.


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