In a triangulated category, cofibre fill-ins always exist. Neeman showed that there is always at least one “good” fill-in, i.e., one whose mapping cone is exact. Verdier constructed a fill-in of a particular form in his proof of the $4\times 4$ lemma, which we call “Verdier good”. We show that for several classes of morphisms of exact triangles, the notions of good and Verdier good agree. We prove a lifting criterion for commutative squares in terms of (Verdier) good fill-ins. Using our results on good fill-ins, we also prove a pasting lemma for homotopy cartesian squares.

中文翻译：

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关于精确三角形的好态射

在三角划分的类别中，cofiber 填充物始终存在。尼曼表明，总是至少有一个“好的”填充，即映射锥是精确的。Verdier 在他的证明中填写了一个特定的表格$4\times 4$引理，我们称之为“Verdier good”。我们表明，对于精确三角形的几类态射，good 和 Verdier good 的概念是一致的。我们根据（Verdier）良好填充证明了可交换平方的提升标准。使用我们对良好填充的结果，我们还证明了同伦笛卡尔平方的粘贴引理。