We will introduce the notion of cellular $\mathbb{A}^1$-homology of a smooth scheme admitting a nice stratification. We will discuss how to orient the cellular $\mathbb{A}^1$-chain complex (whose homology is the cellular $\mathbb{A}^1$-homology) and make computations using it, with focus on the Bruhat decomposition of a split reductive group and the associated flag variety. The talk is based on joint work with Fabien Morel.