
TL;DR
This paper introduces a new categorical framework for Kuranishi spaces, showing they form a 2-category and relate to derived differential geometry, improving their theoretical properties and connections to other structures.
Contribution
The paper proposes a new definition of Kuranishi spaces forming a 2-category, establishing equivalences with existing structures and integrating them into derived differential geometry.
Findings
Kuranishi spaces form a 2-category with better categorical properties.
Any FOOO Kuranishi space can be uniquely associated with a compact Kuranishi space.
Kuranishi spaces are equivalent to derived orbifolds in the context of derived differential geometry.
Abstract
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 -holomorphic curves. We propose a new definition of Kuranishi space, which has the nice property that they form a 2-category . Thus the homotopy category Ho is an ordinary category of Kuranishi spaces. Any Fukaya-Oh-Ohta-Ono (FOOO) Kuranishi space can be made into a compact Kuranishi space uniquely up to equivalence in (that is, up to isomorphism in Ho), and conversely any compact Kuranishi space comes from some (nonunique) FOOO Kuranishi space . So FOOO Kuranishi spaces are equivalent to ours at one level, but our definition has better categorical…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory · Advanced Algebra and Geometry
