Weakly infinite-dimensional spaces modulo simplicial complexes

The classes of spaces ${\mathscr{K}}\text{-}{\rm wid}$ and ${\mathscr{L}}\text{-}{\rm wid}$ are introduced for the class $\mathscr{K}$ of finite simplicial complexes and the class $\mathscr{L}$ of compact polyhedra. If ${\mathscr{K}}={\mathscr{L}}=\{0,1\}$, then ${\mathscr{K}}\text{-}{\rm wid}={\rm wid}$, ${\mathscr{L}}\text{-}{\rm wid}=S\text{-}{\rm wid}$. It is proved that $S\text{-}{\rm wid}\subset{\mathscr{L}} \text{-}{\rm wid}$ and ${\mathscr{L}}\text{-}{\rm wid} =S\text{-}{\mathscr{L}}_\tau\text{-}{\rm wid}$ for any triangulation $\tau$ of the class $\mathscr{L}$.