The Stacks project

Remark 90.10.3. When $\mathcal{F}$ is cofibered in sets, conditions (S1) and (S2) are exactly conditions (H1) and (H2) from Schlessinger's paper [Sch]. Namely, for a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$, conditions (S1) and (S2) state:

  1. If $A_1 \to A$ and $A_2 \to A$ are maps in $\mathcal{C}_\Lambda $ with $A_2 \to A$ surjective, then the induced map $F(A_1 \times _ A A_2) \to F(A_1) \times _{F(A)} F(A_2)$ is surjective.

  2. If $A \to k$ is a map in $\mathcal{C}_\Lambda $, then the induced map $F(A \times _ k k[\epsilon ]) \to F(A) \times _{F(k)} F(k[\epsilon ])$ is bijective.

The injectivity of the map $F(A \times _ k k[\epsilon ]) \to F(A) \times _{F(k)} F(k[\epsilon ])$ comes from the second part of condition (S2) and the fact that morphisms are identities.

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