## 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.