nLab categorical shape theory

In the original form of abstract shape theory, the restrictions that (i) it should refer to a subcategory of ‘nice’ or ‘good’ objects, and (ii) that subcategory should be pro-reflective (shape theorist's 'dense') were unnecessarily restrictive, eliminating interesting situations where similar constructions could be useful, and tended to obscure the nature of certain of the results, which were more much general than was initially apparent.

This was explored by Deleanu and Hilton in the early 1970s.

Definition

Let $K : D\to C$ be a functor. The shape category of $K$ is the category with objects the same objects as those of $C$ but with morphisms from $c$ to $c^\prime$ being the functors between the corresponding comma categories $(c^\prime)/K)$ and $(c/K)$ (note reversal of order) that are compatible with the codomain functors to $D$.

(More to come later!)