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.