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.
- Pub Date:
- August 2020
- Mathematics - Algebraic Topology;
- Mathematics - Category Theory;
- 18G80 (Primary) 55U35 (Secondary)
- v3: Minor changes. Accepted for publication in the Journal of Pure and Applied Algebra