Étale algebraic stacks

Just yesterday, upon some prodding from Michael Thaddeus, I added a two short sections comparing algebraic spaces and algebraic stacks in the fppf and étale topology (see the corresponding sections of chapters Bootstrap and Criteria for Representability). Let me just tell you what the statement is. For algebraic spaces the result is that

If F is a sheaf on (Sch) in the étale topology whose diagonal is representable by schemes and which has an étale covering by a scheme, then F is also a sheaf in the fppf topology hence an algebraic space (as defined in the stacks project).

For algebraic stacks the result is that

If X is a stack in groupoids over (Sch) with the étale topology whose diagonal is representable by algebraic spaces and which has a smooth covering by a scheme, then X is also a stack for the fppf topology hence an algebraic stack (as defined in the stacks project).

Till yesterday I had filed away this material under the heading: “Things that have to go into the stacks project at some point but which are not as interesting as other material I am working on now.” However, I probably should have worked it out sooner as some related remarks in the stacks project were misleading (I have now removed these remarks, see the red text in this commit).

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