On good morphisms of exact triangles

TL;DR

This paper investigates the properties of good morphisms of exact triangles in triangulated categories, establishing conditions under which different notions of goodness coincide and deriving key lemmas for homotopy cartesian squares.

Contribution

It characterizes when good and Verdier good fill-ins agree and provides a lifting criterion and pasting lemma for homotopy cartesian squares in triangulated categories.

Findings

Good and Verdier good fill-ins coincide for several classes of morphisms.

A lifting criterion for commutative squares using good fill-ins is established.

A pasting lemma for homotopy cartesian squares is proved.

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.