## Invariants of spaces over some metric space

We consider the category of locally connected topological spaces
over some complete metric space \(M\), whose objects are continuous
functions to \(M\) and whose morphisms between two given functions
\(f \colon X \rightarrow M\) and \(g \colon Y \rightarrow M\) are
the continuous maps \(\varphi \colon X \rightarrow Y\) such that \[
\xymatrix{
X \ar[rr]^{\varphi} \ar[dr]_f & & Y \ar[dl]^g \\
& M
}
\] commutes. In the following we will consider several invariants
(mostly given as functors to other categories and mostly in the
special case where \(M = {\mathbb{R}}\)) under isomorphisms of
objects in this category.