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atlas on X, as in [42, Def. 2.3.1], if it also satisfies:
• Let V = (V0 ⊃ V1 ⊃ · · · ) be an scBanach space and U ⊆ V0 be open.
(O, V ) and a homeomorphism ψ : O → Im ψ with an open set Im ψ ⊂ Z. • Mpolyfold charts (O, V , ψ), (O˜, V˜, ψ˜) on Z are compatible if ψ˜−1 ◦ ψ ◦ r :
U → V˜ and ψ−1 ◦ ψ˜ ◦ r˜ : U˜ → V are scsmooth, where U ⊂ V0, U˜ ⊂ V˜0 are
and Im r = ψ−1(Im ψ˜) ⊆ O, Im r˜ = ψ˜−1(Im ψ) ⊆ O˜. • An Mpolyfold is roughly a metrizable topological space Z with a maximal
atlas of pairwise compatible Mpolyfold charts. That is, composition of 1morphisms is associative up to canonical 2isomorph
ism, as for weak 2categories in §A.1.
If g : X → Z and h : Y → Z are 1morphisms in Kur with g a w
then g, h are strongly transverse. Thus Theorem 5.7 implies:
Now let f : Y1 → Y2 be a smooth map of manifolds. Define the pushforward