  We develop a theory of perfect algebraic stacks that extend our theory of
perfect algebraic spaces in arXiv:2303.07672, arXiv:2303.08502 to the setting
of algebraic stacks. We prove several desired properties of perfect algebraic
stacks. This extends some previous results of perfect schemes and perfect
algebraic spaces, including the recent one developed by Bertapelle et al. in
arXiv:1611.02060. Moreover, our theory extends the previous one developed by
Xinwen Zhu in arXiv:1707.05700.
  Our method to define perfect algebraic stacks differs from all previous
approaches, as we utilize representability of algebraic spaces. There is a
natural notion of algebraic Frobenius morphisms of algebraic stacks. The
algebraic Frobenius morphism provides one with an explicit description of the
perfection of an algebraic stack. This gives rise to the perfection functor on
algebraic stacks, which enables us to pass between the usual and the perfect
world.
