Homotopy cartesian squares in extriangulated categories

TL;DR

This paper extends the concept of homotopy cartesian squares to extriangulated categories, showing that the composition of two such squares remains homotopy cartesian, generalizing known results from abelian and triangulated categories.

Contribution

It introduces and proves a property of homotopy cartesian squares within extriangulated categories, broadening the theoretical framework beyond classical categories.

Findings

Composition of two homotopy cartesian squares is homotopy cartesian in extriangulated categories.

Generalizes Mac Lane's result for abelian categories.

Extends Christensen and Frankland's result for triangulated categories.

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category. Given a composition of two commutative squares in $\mathcal{C}$, if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian. This covers the result by Mac Lane (1998) for abelian categories and the result by Christensen and Frankland (2022) for triangulated categories.