Suppose $\kappa\colon\mathcal{Q}\rightarrow\mathcal{P}$ is an order preserving, non-rank-increasing correspondence between the graded posets $\mathcal{Q}$ and $\mathcal{P}$. Define the barycentric subdivision $\operatorname{bsd}(\mathcal{Q},\kappa)$ of $\mathcal{Q}$ with respect to $\kappa$ as the subposet

$$\{(p\ ,\ q_{0}\prec q_{1}\prec\ldots\prec q_{r})\in\mathcal{P}\times\operatorname{bsd}(\mathcal{Q})\ :\ p\preceq\kappa(q_{0})\}$$

of the product poset $\operatorname{bsd}(\mathcal{Q})\times\mathcal{P}$.