  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.
