Beck-Chevalley Condition

Let ℰ → ℬ → ℰ ℬ \mathcal{E}\to\mathcal{B} be a Cartesian Fibration, and let f ⁢ g = h ⁢ k 𝑓 𝑔 ℎ 𝑘 fg=hk be a square of morphisms in ℬ ℬ \mathcal{B} . Moreover, suppose that we have Left Adjoints L g : ℰ a → ℰ b : subscript 𝐿 𝑔 → subscript ℰ 𝑎 subscript ℰ 𝑏 L_{g}:\mathcal{E}_{a}\to\mathcal{E}_{b} , L h : ℰ c → ℰ d : subscript 𝐿 ℎ → subscript ℰ 𝑐 subscript ℰ 𝑑 L_{h}:\mathcal{E}_{c}\to\mathcal{E}_{d} to the base change functors g ∗ : ℰ b → ℰ a : superscript 𝑔 → subscript ℰ 𝑏 subscript ℰ 𝑎 g^{*}:\mathcal{E}_{b}\to\mathcal{E}_{a} , h : ℰ d → ℰ c : ℎ → subscript ℰ 𝑑 subscript ℰ 𝑐 h:\mathcal{E}_{d}\to\mathcal{E}_{c} , resp.

The Beck-Chevalley condition

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