Affiliation:
1. College of Science, Hunan University of Technology and Business , 410205 Changsha , P. R. China
2. College of Mathematics, Hunan Institute of Science and Technology , 414006 Yueyang , Hunan , P. R. China
Abstract
Abstract
Let
(
C
,
E
,
s
)
\left({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}})
be an extriangulated category. Given a composition of two commutative squares in
C
{\mathcal{C}}
, if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by Mac Lane [Categories for the Working Mathematician, Second edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998] for abelian categories and by Christensen and Frankland [On good morphisms of exact triangles, J. Pure Appl. Algebra 226 (2022), no. 3, 106846] for triangulated categories.
Reference13 articles.
1. A. Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. 9 (1957), no. 2, 119–221, DOI: https://doi.org/10.2748/tmj/1178244839.
2. J. Verdier, Des catégories dérivées des catégories abéliennes, With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis, Astérisque No. 239, 1996.
3. B. J. Parshall and L. L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton University Mathematical Notes 3 (1988), 1–104.
4. A. Neeman, Triangulated categories, Annals of Mathematics Studies, Vol. 148, Princeton University Press, Princeton, 2001.
5. H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég. 60 (2019), no. 2, 117–193.