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arXiv · 2008.03643

On good morphisms of exact triangles

J. Daniel Christensen·Martin Frankland
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Abstract

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.

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J. Daniel Christensen, Martin Frankland. 2020-08-09. On good morphisms of exact triangles. https://doi.org/10.1016/j.jpaa.2021.106846

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