Today I was able to add the result that the quotient stack [U/R] associated to a smooth groupoid in algebraic spaces is an algebraic stack. See Theorem Tag 04TK. Very satisfying!

There are several things that have to be done next:

  1. Work on the chapter Examples of Stacks, and start a parallel chapter Examples of Algebraic Stacks, where we discuss in detail some very basic examples of algebraic stacks.
  2. Add some foundational material on changing the base scheme and the underlying big site. This could then be used to define a 2-category of algebraic stacks which lumps all algebraic stacks together regardless of big site that was used to define them (but I’m not sure this would be useful, so I may not add this).
  3. Write a Chapter on properties of algebraic stacks
  4. Write a Chapter on morphisms of algebraic stacks
  5. Write about separation axioms for morphisms of algebraic stacks
  6. And so on and so forth.

Next week I will not have time to work on the stacks project, so you can start working on the topics above yourself!

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