atlas on X, as in [42, Def. 2.3.1], if it also satisfies:
• Let V = (V0 ⊃ V1 ⊃ · · · ) be an sc-Banach space and U ⊆ V0 be open.
(O, V ) and a homeomorphism ψ : O → Im ψ with an open set Im ψ ⊂ Z. • M-polyfold 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 sc-smooth, where U ⊂ V0, U˜ ⊂ V˜0 are
and Im r = ψ−1(Im ψ˜) ⊆ O, Im r˜ = ψ˜−1(Im ψ) ⊆ O˜. • An M-polyfold is roughly a metrizable topological space Z with a maximal
atlas of pairwise compatible M-polyfold charts. That is, composition of 1-morphisms is associative up to canonical 2-isomorph-
ism, as for weak 2-categories in §A.1.
If g : X → Z and h : Y → Z are 1-morphisms 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
f∗ : M Ck(Y1; R) → M Ck(Y2; R) for k ∈ Z to be the unique R-linear map defined
on generators [V, n, s, t] of M Ck(Y1; R) by
There are obvious notions of composition G ◦ F of strict and weak 2-functors
F : C → D, G : D → E, identity 2-functors idC, and so on.
If C, D are strict 2-categories, then a strict 2-functor F : C → D can be