On groups and groupoids (in the nontechnical sense)

At this point I’ve put almost everyone who wanted to be in a group (and who hasn’t just signed up) into a group, and invited everyone to zulip and discord.  If you don’t know if you are in a group, you can go to zulip, and see if you are in any of the streams for any groups.  (I may have just added you without telling you.)  Also, if you’d like to join a particular group, anyone in that group can just subscribe you to that stream — please go ahead and do that.

I’ve found that making groups, and figuring out how to put people into groups, takes far more time than I’d expected.  (In fact, of all the things in this pseudocourse, this, of all things, was the task that was most onerous — not what I was expecting.)  So from now on, if you are not in a group (e.g. if you’ve just joined, and have gotten your invitation to zulip) and you want to be, feel free to ask me, but also feel free to ask around (perhaps people you know, perhaps just describe your background in one of the groupoids) and see if one of the groups might just take you in.

Also, you can certainly be part of more than one group if that suits you.

And now may be a good time to start rationalizing groups — if you are in a group that is fairly quiet (or completely quiet), then you can jump groups, or we can even fold the group into another one.  There are no hard and fast rules here — we just want to do whatever works well, and whatever makes people feel comfortable getting into conversations.

On stacks (in the technical sense)

A number of people are interested in hearing more about stacks, and I’m definitely open to it, with a number of caveats that are predictable to the experts.  But Taylor Dupuy  mentioned some things in the Groupoid D stream on zulip that I didn’t know about, and wanted to advertise here.

First, quoting Taylor,

before diving into the technicalities of what an algebraic stack is, you should probably read DZB’s advice here: http://www.math.emory.edu/~dzb/adviceStacks.html.

(DZB = David Zureick-Brown)

Second, Taylor has actually explained a bunch of things related to stacks!

Here is a playlist on Grothendieck Topologies:

https://www.youtube.com/playlist?list=PLJmfLfPx1Oed3osC36YKSZHJbvUWUjQ2m

Here is a video on Stacks and descent data (abstract gluing data):
https://www.youtube.com/watch?v=91fJ3GTM7Dk&t=807s

Here is a videos on morphisms of Fibered Categories (you need fibered categories to talk about stacks):
https://www.youtube.com/watch?v=piS-9sz7fkI

Here is a video on Gerbes:
https://www.youtube.com/watch?v=4sv40lsj0s4&list=PLJmfLfPx1Oec9YzTuiC-huiAGPEES3YN6

Here is a video on the idea of Algebraic Stacks:
https://www.youtube.com/watch?v=9SrNfj5OE8s&list=PLJmfLfPx1Oec9YzTuiC-huiAGPEES3YN6&index=3

Here is a video on the idea behind Algebraic Spaces and Stacks (and the representability issues) [This is where things get hard IMHO):
https://www.youtube.com/watch?v=F_-lS-pn5pQ

Here is a video on representability of Morphisms:
https://www.youtube.com/watch?v=FtHHK_sLZSg

Here is a video on stackification:
https://www.youtube.com/watch?v=0c152d66FUI&list=PLJmfLfPx1Oec9YzTuiC-huiAGPEES3YN6&index=7

Here is a paper by Moerdijk which I think is the best introduction to Stacks which a lot of those videos is based on: https://arxiv.org/pdf/math/0212266.pdf (I also used Moret-Bailey, Olssen’s book (which is the best book nowadays), and the Stacks Project). I also watched people like DZB, Ravi, and Max Leiblich talk a lot as a grad student so my videos are just like me copying them poorly.

I have some other videos on band of Gerbes, but I don’t think that is really important. In fact, I don’t think much of this is really that important for the first time through. I think you should be looking at more basic examples like curves, surfaces, linear series etc. Also, before learning about algebraic stacks you should figure out what an analytic stack (orbifold is) and why they matter. (IMHO: I think you should wait until you actually need something before you start to learn it, otherwise you run the risk of drowning in papers you don’t understand—that is what happens to me at least. Also, I forget it if I don’t use it!). Don’t worry… trouble will find you… you don’t need to go looking for it.

Not surprisingly, I agree with his point of view.  (Not surprisingly, I wanted Taylor and DZB to be shepherds….)

Something else from Taylor in Groupoid D, at nearly the same time, that I want to remember:

Commutative algebra is where the hard parts of algebraic geometry go to hide.

– Christelle Vincent

Thought of the day

While watching Abi Ward’s successful Ph.D. defense today (congratulations Abi!), I had an epiphany.  There are a few adjectives to the noun “functor” — full, faithful, essentially surjective, equivalence.  I now realize these different-sounding names are hiding their sameness.  In retrospect, essential surjective should be 0-surjective, full should be 1-surjective, and faithful should be 2-surjective.  And equivalence = 0-surjective  + 1-surjective + 2-surjective.  Then “generalizing downwards”, if you have a map of sets, then a surjective map of sets could be called “0-surjective”, and an injective map of sets would be “1-surjective”, and “0-surjective + 1-surjective = bijective.”.    In between, if you have a morphism in a category, 0-surjective would be “epimorphism”, and 1-surjective would be “monomorphism”, and 0-surjective + 1-surjectve would be “isomorphism”.

I should explain why this is true.  Certainly I am happy that I knew all of these different words before I’d try to understand them in terms of very few words.  (Similarly in a category, we could replace “object” by “0-morphism”, “morphism in a category”  or “functor between categories” by “1-morphism”, and “natural transformation of functors” by “2-morphism”…  then we could do away with almost all of these words, and just have numbers, “surjective” and “morphism”.  But then our heads would explode.