Hi Robert, the \$$G\$$ you propose to build in [#2](https://forum.azimuthproject.org/discussion/comment/18957/#Comment_18957) is more easily said than done but is very interesting. The good way to do it is fastly viewed in this Spivak's [slides](http://math.mit.edu/~dspivak/informatics/talks/CTDBIntroductoryTalk) where he spells the Grothendiek construction applied in this case. Your \$$G\$$ is his \$$\pi:Gr(I) \to \mathcal{C}\$$ of slide 53. What a pity that this seems not to be included in the book! Perhaps we can talk about it here a bit.

This idea allows one to describe catgorical intercourse between different strata of the type hierarchy!