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I am planning to write a few things about Picard groupoids. For this purpose, I have removed a couple of redirects from Picard 2-group, added a new one which is a bit more precise, and tweaked the beginning of this page slightly. Feel free to edit further; I basically just wished to free up the page Picard groupoid.
I am also attempting to tweak all existing links to Picard groupoid appropriately.
Hopefully now done: Deligne’s theorem on tensor categories, Picard scheme, geometry of physics – superalgebra, super vector space, invertible object. At the latter, I removed the reference to Picard groupoid, as I wish to reserve this for a slightly different notion. If someone objects, just raise it, I am sure we can find some solution; for now, I am just clearing the way to actually write something at Picard groupoid which does not contradict an earlier reference.
Made a beginning at Picard groupoid now.
Added a mention that they model stable homotopy 1-types. Obviously references and a proof could be given. I would be interested to know if the invertibility criterion can be taken to be strict (it is known that everything else can be).
Added a paragraph to symmetric monoidal 2-category to give the strictest possible definition (which is very easy, of course).
At Picard groupoid, I have now added a few 2-categorical considerations which show that the homotopy category of the category of Picard groupoids is additive. I have written this part rather concisely. In particular, the facts that I use, which are very basic 2-category theory, do not appear as far as I know on the nLab; it would be nice if someone would like to add them. Anyone is very welcome to elaborate on what I have written.
What I am actually interested in here is outlining a proof that the homotopy category of Picard groupoids is triangulated, and in fact moreover ’4-angulated’, in the sense of the higher triangulated categories of Maltsiniotis. These observations are, as far as I know, new. I will probably transfer them to some kind of paper if they hold up, but for now I’ll probably work here, off and on.
Could you say again which changes you are referring to in #3? I checked super vector space and some others, and the system does not show any changes to the text. Did you change redirects at these entries?
Ah, sorry, I thought that it would show up in the diff. Basically nothing should be changed at all from a reader’s point of view: no text should be changed, and the page which ’Picard groupoid’ links to at super vector space etc should still be the same. Because I removed a redirect for ’Picard groupoid’ from Picard 2-group, and replaced it by a redirect for ’Picard groupoid of a monoidal category’, it was necessary for me to change ’Picard groupoid” with square brackets around it to ’Picard groupoid of a monoidal category|Picard groupoid’ with square brackets around it at some pages.
The only case where I have changed a little content is at invertible object and Picard 2-group.
Thanks, now I understand what you are doing. That’s good. But so I have added a disambiguation line at the beginning of the new Picard groupoid, saying
This page considers Picard groupoids in themselves. For the concept of Picard groupoid in a monoidal category, see there.
Also I added at the bottom a “Related concept” seciton with a pointer to 2-group
Nice, thanks!
I suppose we should still comment on the fact that some people take “Picard groupoid” to imply symmetry (as on your page) and others do not (as on mine).
This inconsistncy seems to pervade the literature. For instance in this MO discussion the first answer takes it without symmetry, the second with symmetry.
That’s okay, we can’t change this. But let’s add comments to the pages highlighting the issue.
(I won’t do any further editing right now. I need to call it quits.)
I also will not do further editing tonight. Yes, adding remarks sounds like a good plan. I am fine with attempting to establish some terminology; say ’abelian Picard groupoid’ or ’symmetric Picard groupoid’ for the notion I am writing of. Just let me know (or feel free to edit it in).
Yes, please add some remarks like this. Also regarding the strictness that you impose.
Formally, we can obtain it by truncating? the 2-category of Picard groupoids, namely identifying all 1-arrows which are 2-isomorphic, and throwing away the 2-arrows.
I changed truncating to decategorifying.
Thanks David! I have added a couple of small sections at decategorification. There were are a couple of typos in redirects at Ho(Top), which I corrected; I hope that this does not break anything (I am assuming that nobody has used these redirects, since presumably they would in that case have corrected them).
I have not yet found time to add the remarks that you ask for, Urs. If anyone wishes to do so in the meantime, they are welcome.
Thanks for catching this, I have fixed it now to “…core of its full subcategory…”.
It is true that a category with groupal monoidal structure is automatically a groupoid. But if we pass to the full subcategory on the tensor-invertible objects in any monoidal category, the result is not yet a groupal monoidal category, and we need to further restrict to the invertible morphisms to make it one.
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