and including precise formulations of definitions and tools/results and quoting locations of origins. Readers should not let technical details in certain sections deter them and should read relevant virtual theories that interest them. Compatibility of various existing gluing methods is not discussed here and might be pursued in a future article.

I must admit that I have not thoroughly investigated the entire package of each approach, and I mostly get informed by listening to talks by or through personal discussions with one/two member(s) of each group. The exceptions are Hofer-Wysocki-Zehnder’s polyfold package for Gromov-Witten theory and Fukaya-Oh-Ohta-Ono’s Lagrangian Floer theory in [@FOOO] (sans the gluing analysis part of the latter) that I have been looking at more closely and thinking about for quite a while. The identification of the structures and compatibility of the perturbation theories between HWZ’s sc-Fredholm structure (or interchangeably, polyfold Fredholm structure) in polyfold theory and FOOO’s Kuranishi structure imposed with two extra natural conditions \[MAXMATCHING\] (1) and (3) are the topics of my thesis [@Dingyu], and they are being improved and expanded as a series of papers [@DI], [@DII] and [@DIII], the latter two of which should be available in next few months. We remark that 