publication . Preprint . 2015

Kuranishi spaces as a 2-category

Joyce, Dominic;
Open Access English
  • Published: 26 Oct 2015
This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli spaces of $J$-holomorphic curves. We propose a new definition of Kuranishi space, which has the nice property that they form a 2-category $\bf Kur$. Thus the homotopy category Ho$({\bf Kur})$ is an ordinary category of Kuranishi spaces. Any Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space $\bf X$ can be made into a compact Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that is, up to isomorphism in Ho$({\bf Kur})$...
arXiv: Mathematics::Symplectic GeometryMathematics::Algebraic GeometryMathematics::Differential GeometryMathematics::Complex Variables
free text keywords: Mathematics - Symplectic Geometry, Mathematics - Differential Geometry
Funded by
RCUK| Lagrangian Floer cohomology and Khovanov homology
  • Funder: Research Council UK (RCUK)
  • Project Code: EP/H035303/1
  • Funding stream: EPSRC
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41 references, page 1 of 3

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

41 references, page 1 of 3
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