The spring semester has just ended here at Columbia. I assume for most of you similarly the summer will start soonish. Besides running an REU, supervising undergraduates, talking to graduate students, going to conferences, and doing research, I plan to write about Artin’s axioms for algebraic spaces and algebraic stacks this summer.

What are your summer plans? Maybe you intend to work through some commutative algebra or algebraic geometry topic over the summer. In that case take a look at the exposition of this material in the stacks project (if it is in there) and see if you can improve it. Or, if it isn’t there, write it up and send it over. If you want to coordinate with me, feel free to send me an email.

Another (more nerdy?) project is to devise an online comment system for the stacks project. It could be something simple (like a comment box on the lookup page) or something more serious such as a wiki, blog, or bug-tracker, etc. If you are interested in creating something (and you have the skills to do it), please contact me about it.

Finally, I’ve wondered about having mirrors of the stacks project in (very) different locations. If you know what this entails and you are interested in running one, please contact me.

One thing which might be nice for the stacks project is the proof that the 2-category of foo-stacks is the localisation of the 2-category of blah-groupoids, for appropriate values of foo and blah. Dorette Pronk has a result in this direction, and I am aiming to expand her result to cover a wide range of definitions of algebraic stacks, and other sorts of stacks. The proof however relies on a theorem whose proof is heavy on 2-categorical (abstract nonsense) calculations. If you are willing to take that on faith, then I see it as a potentially useful addition.

In the stacks project there is no such thing as taking things on faith. Also when you say foo, people are next going to expect you to say bar…

Seriously though, we have “stackyfication” described in the chapter on stacks. Sure, it is somewhat rough and the exposition and proofs could be improved (this would be very helpful). Adding a remark on what it means to iocalize a 2-category and then adding a remark on how the process of stackyfication is one of these localizations would be very helpful too. Since in algebraic geometry we usually are interested in a finite list of particular algebraic stacks (such as M_g and A_g) and their interrelationships, this should be enough for now…

A similar discussion can be had about the relationship between the 2-category of “presentations” (i.e., smooth groupoids (U, R, s, t, c) in algebraic spaces with suitable notions of 1-morphisms and 2-morphisms) and the 2-category of algebraic stacks. I’ve always thought that it is enough to show that every stack of the form [U/R] is algebraic and conversely that every algebraic stack is of the form [U/R]. Others find it useful to compare the 2-categories. The problem with doing this is that this leads into 3-category land which we try to avoid at all cost.

Namely, my feeling is that in the stacks project we should avoid doing explicit 3-category things, until we decide it is really necessary, and then we go all the way and do infinity categories.

>with suitable notions of 1-morphisms and 2-morphisms

and that is exactly the point. Saying what these 1- and 2-morphisms are, and why they form a (possibly weak) 2-category is not trivial.

>I’ve always thought that it is enough to show…

only if you think essential surjectivity is enough to show a pair of 2-categories are equivalent 😉 It is possible to show a pair of 2-categories are equivalent without descending into 3-categories.

I’ll have a read of the stackification section.

I think you misunderstood my comment “I’ve always…”. What I meant is that it is enough for most applications of the notion of a presentation of an algebraic stack in algebraic geometry.

Ah, yes, that makes sense. Coming from category theory, I want the whole story, 2-categories and all…

Sorry to keep going on about this, but Desirables, section 12 (http://math.columbia.edu/algebraic_geometry/stacks-git/desirables.pdf) asks for exactly what the suitable 1-morphisms are. I can help you there, even if you don’t want to go into detail about localisation of 2-categories.

OK, that is a good idea! The chapter “Groupoids in Algebraic Spaces” has some material like this already (but not as far as I can see the answer to exactly this question).

I should mention though that unfortunately, the desirables chapter is no longer up to date and needs to be updated.