Symmetric semi-perfect obstruction theory revisited

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne-Mumford stack $X$ constructed by Chang-Li, and Behrend’s theorem equating the weighted Euler characteristic of $X$ and the virtual count of $X$ by symmetric semi-perfect obstruction theories. As an application, we prove that Joyce’s d-critical scheme admits a symmetric semi-perfect obstruction theory, which can be applied to the virtual Euler characteristics by Jiang-Thomas.