Splendid find, Jesus, thank you! I'm afraid I never went deep enough into the woods to physically meet a construction named after Grothendieck! But now that you've invoked the master, it can't be long until we get the summoned help...
Page 51 may be a good entry point compare Spivak's slides. However, looking at the diagrams on that page I'm having trouble to understand the difference between Gr(I) and I. Mostly because of the label on the arrow (b1,b1,b2). Because of that it really looks as if the diagram I had all the info Gr(I) has.
Be that as it may, I like the intuition that π collapses messy reality to a clean model. Just like what you do when trying to find out what to solve (and how).
**Puzzle rf4:** I suppose _Strata_ means layers. What lies above C? Are there any more below Gr(I)?