Fine shape III: $\mathrm{\Delta }$-spaces and $\mathrm{\nabla }$-spaces

In this paper we obtain results indicating that fine shape is tractable and “not too strong” even in the non-locally compact case, and can be used to better understand infinite-dimensional metrizable spaces and their homology theories. We show that every Polish space $X$ is fine shape equivalent to the limit of an inverse sequence of simplicial maps between metric simplicial complexes. A deeper result is that if $X$ is locally finite dimensional, then the simplicial maps can be chosen to be non-degenerate. They cannot be chosen to be non-degenerate if $X$ is the Taylor compactum.