instIsStableUnderCompositionJointlySurjectivePrecoverage

English

If pullback along f and g preserve surjectivity of the canonical map, then the comap of jointlySurjectivePrecoverage is stable under base change.

Русский

Если каноническое отображение сохраняет сюръективность при тягке по f и g, то comap jointlySurjectivePrecoverage устойчиво к базовым изменениям.

LaTeX

$$$(H \colon \forall \{X,Y,S\}, (f:X\to S)(g:Y\to S) [HasPullback f g], Surjective( pullbackComparison F f g)) \Rightarrow \text{IsStableUnderBaseChange}$$$

Lean4

instance : jointlySurjectivePrecoverage.IsStableUnderComposition where
  comp_mem_coverings {ι} S X f hf σ Y g
    hg := by
    simp_rw [ofArrows_mem_jointlySurjectivePrecoverage_iff] at hf hg ⊢
    intro x
    obtain ⟨i, y, rfl⟩ := hf x
    obtain ⟨j, z, rfl⟩ := hg i y
    use ⟨i, j⟩, z
    simp

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