Polyhedral products in abstract and motivic homotopy theory

Abstract: We introduce polyhedral products in an $\infty$-categorical setting. We generalize a splitting result by Bahri, Bendersky, Cohen, and Gitler that determines the stable homotopy type of the a polyhedral product. We also introduce a motivic refinement of moment-angle complexes and use the splitting result to compute cellular $\mathbb{A}^1$-homology, and $\mathbb{A}^1$-Euler characteristics.