The Stacks project

LaTeX: \Ob

	Introduced a macro

	\def\Ob{\mathop{\rm Ob}\nolimits}

	and replaced any occurence of \text{Ob}( with \Ob(. There are
	still some occurences of \text{Ob} but these are sets, not the
	operator that takes the set of objects of a category.

LaTeX: \Sch

	Introduced a new macro

	\def\Sch{\textit{Sch}}

	and replaced all the occurences of \textit{Sch} with \Sch.

Reduced algebraic stacks

Reduced algebraic stacks

Definition of a presentation of an algebraic stack

	Plus some small local improvements.

Added new tags

2-fibre products of algebraic stacks

	This looks complicated because of or insistence that in the
	first chapter on algebraic stacks we use a notation which
	distinguishes between schemes, algebraic spaces, and algebraic
	stacks.

More cleaning up of Algebraic Stacks

	More or less OK proof of the easy direction for going between
	algebraic stacks and presentations. It would be good to add more
	material allowing us to work more easily with fibred
	categories...

Every algebraic stack has a presentation

	This is the easy direction, although there is plenty to improve
	on here. We should split out some of the discussion in separate
	lemmas. In particular, we should have a discussion on criteria
	which garantee that a morphism of stacks in groupoids is an
	equivalence. We should discuss more generally the construction
	where given
		U ---> X
	with X an algebraic stack, U an algebraic space, on setting R =
	U \times_X U we get a morphism of stacks in groupoids
		[U/R] ---> X
	for free. Then the lemma becomes much more readable and just
	says that if U ---> X is smooth and surjective, then the
	associated 1-morphism is an equivalence.