This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that have trivial shape in the classic Borsuk sense (and thus also in the sense of Cech homology). However, these spaces can be represented as inverse limits of finite trees, when considered as topological spaces. In particular, my spaces are all finite-dimensional, connected, compact metric spaces and have trivial shape in the classical sense; they are also path-connected, and simply connected, though not necessarily contractible.
So what I'm looking for is something like a shape operator or geometric-combinatorial equivalence class which doesn't necessarily preserve trivial shape with respect to inverse limits, and yet is sensible for non-locally connected spaces (as mine generally are).
Are there any such theories that spring to mind?