Kuranishi spaces as a 2-category

Abstract

This is a survey of the author's in-progress book arXiv:1409.6908. '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$({\bf Kur})$), and conversely any compact Kuranishi space $\bf X'$ comes from some (nonunique) FOOO Kuranishi space $\bf X$. So FOOO Kuranishi spaces are equivalent to ours at one level, but our definition has better categorical properties. The same holds for McDuff and Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. Using results of Yang on polyfolds and Kuranishi spaces surveyed in arXiv:1510.06849, a compact topological space $X$ with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185) can be made into a Kuranishi space $\bf X$ uniquely up to equivalence in $\bf Kur$. Our Kuranishi spaces are based on the author's theory of Derived Differential Geometry (see e.g. arXiv:1206.4207), the study of classes of derived manifolds and orbifolds that we call 'd-manifolds' and 'd-orbifolds'. There is an equivalence of 2-categories ${\bf Kur}\simeq{\bf dOrb}$, where $\bf dOrb$ is the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived orbifold. We discuss the differential geometry of Kuranishi spaces, and the author's programme for applying these ideas in symplectic geometry.