Sectional category of maps related to finite spaces

In this study, we compute some examples of sectional category secat$(f)$ and sectional number sec$(f) for continuous maps $f$ related to finite spaces. Moreover, we introduce an invariant secat$_k(f)$ for a map $f$ between finite spaces using the $k$-th barycentric subdivision and show the equality secat$_k(f)=$ secat$(B(f))$ for sufficiently large $k$, where $B(f)$ is the induced map on the associated polyhedra.