Idempotent completion of cubes in posets

Abstract

This note concerns the category $\Box$ of cartesian cubes with connections, equivalently the full subcategory of posets on objects $[1]^n$ with $n \geq 0$. We show that the idempotent completion of $\Box$ consists of finite complete posets. It follows that cubical sets, ie presheaves over $\Box$, are equivalent to presheaves over finite complete posets. This yields an alternative exposition of a result by Kapulkin and Voevodsky that simplicial sets form a subtopos of cubical sets.