I have finally proved the induction step for the construction of a Fréchet manifold that is a limit of a large diagram of such manifolds, by a very carefully chosen iterated sequence of pullbacks of submersions. The base case requires one to construct by hand, in the notation of the picture below, the manifolds for .

Here is a finite set, with elements , indexing the sets in a closed cover of a compact manifold , and the subsets etc index ‘partial subcovers’. That is, the closed sets corresponding to the elements of that subset, which do not themselves form a cover. Let . The manifolds are a priori the diffeological spaces of functors , where is a fixed finite-dimensional Lie groupoid, and here is the diffeological Cech groupoid of . The aim here is to show that is in fact a Fréchet manifold, by induction on the size of . This result is Proposition 5 of my MATRIX Annals note with Raymond Vozzo, *The smooth Hom-stack of an orbifold* (publisher link, arXiv), with the proof there only stating

*This* diffeological space is what we show is a Fréchet manifold, by **carefully writing the limit as an iterated pullback of diagrams** involving maps that are guaranteed to be submersions by Proposition 3 and Theorem 4, and using the fact that X is appropriately coskeletal

“Carefully writing the limit” indeed! Past me was highly constrained by page limits…