Representing fine shape of local compacta by homotopy classes of ordinary maps

Fine shape, as defined by Melikhov, is an extension of the strong shape category of compacta (compact metrizable topological spaces) to all metrizable spaces, notable for being compatible with both Čech cohomology and Steenrod-Sitnikov homology. In this work we study fine shape of local compacta (locally compact separable metrizable spaces), and construct, for every local compactum $X$, a space $\mathrm{\mid }X\mathrm{\mid }$ unique up to a homotopy equivalence and such that fine shape classes from any locally compact metrizable space $Y$ to $X$ bijectively correspond to homotopy classes of ordinary maps from $Y$ to $\mathrm{\mid }X\mathrm{\mid }$. This correspondence is (contravariatly) functorial in $Y$, thus giving a representation of $Y$-dependent contravariant functor for a fixed $X$; the universal class corresponding to the identity map of $X$ is the homotopy class of a specific embedding of $X$ into $\mathrm{\mid }X\mathrm{\mid }$ that is a fine shape equivalence.