draft-mo-answer.txt

https://math.stackexchange.com/questions/2302051/the-classifying-topos-of-local-rings-and-algebraic-geometry

* [Deligne's theorem](http://stacks.math.columbia.edu/tag/06VX) ensures that
  all the toposes which are in use in algebraic geometry have enough points,
  such that, for instance, for checking whether a morphism of sheaves is an
  epimorphism it suffices to check this on stalks.
  
  However, this theorem is only really useful in conjunction with knowledge how
  the points look like. If the theory which a topos classifies is known, the
  description of the points can be directly read off. (They're just the models
  of the theory in the category of sets.)

* Knowing the points can help in proving that pushing forward along
  closed immersions is exact. See for instance [Gabber and Kelly's
  article](https://arxiv.org/abs/1407.5782).

* Good for specifying maps between toposes?

* Good for knowing which axioms are satisfied internally.
  (In turn useful for constructive or short proofs.)
  (But mention phenomenon of nongeometric sequents.)

* Constructive definitions

* Meta: Look at Wraith.

* Conceptual clarity.