Homotopy cartesian squares in extriangulated categories. (English) Zbl 1527.18014
The notion of homotopy cartesian square first appeared in the study by B. J. Parshall and L. L. Scott [in: Algebra, Proc. Workshop, Ottawa/Can. and Moosonee/Can. 1987, Math. Lect. Note Ser., Expo. Math., CRAF, Carleton Univ. 3, 105 p. (1988; Zbl 0711.18002)]. In this paper under review, the authors study homotopy cartesian squares in extriangulated categories.
Let \((\mathcal{C},\mathbb{E},\mathfrak{s})\) be an extriangulated category. Given a composition of two commutative squares in \(\mathcal{C}\), the authors showed that if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by S. Mac Lane [Categories for the working mathematician. Cham: Springer (1971; Zbl 0232.18001)] for abelian categories and by J. D. Christensen and M. Frankland [J. Pure Appl. Algebra 226, No. 3, Article ID 106846, 33 p. (2022; Zbl 1475.18018)] for triangulated categories.
Let \((\mathcal{C},\mathbb{E},\mathfrak{s})\) be an extriangulated category. Given a composition of two commutative squares in \(\mathcal{C}\), the authors showed that if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by S. Mac Lane [Categories for the working mathematician. Cham: Springer (1971; Zbl 0232.18001)] for abelian categories and by J. D. Christensen and M. Frankland [J. Pure Appl. Algebra 226, No. 3, Article ID 106846, 33 p. (2022; Zbl 1475.18018)] for triangulated categories.
Reviewer: Yonggang Hu (Beijing)
MSC:
| 18G80 | Derived categories, triangulated categories |
| 18E10 | Abelian categories, Grothendieck categories |
Keywords:
extriangulated category; homotopy cartesian square; triangulated category; abelian category; push-out and pull-backReferences:
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