On good morphisms of exact triangles
Abstract
In a triangulated category, cofibre fillins always exist. Neeman showed that there is always at least one "good" fillin, i.e., one whose mapping cone is exact. Verdier constructed a fillin 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 fillins. Using our results on good fillins, we also prove a pasting lemma for homotopy cartesian squares.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.03643
 Bibcode:
 2020arXiv200803643C
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 18G80 (Primary) 55U35 (Secondary)
 EPrint:
 v3: Minor changes. Accepted for publication in the Journal of Pure and Applied Algebra