Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
just a minimum for the moment, in order to record the definition in:
Aleks Kissinger has contacted me about his aims to start a collection of nLab entries on quantum information from the point of view of the Bob Coecke school.
Being very much delighted about this offer, I created a template entry quantum information for his convenience.
created effects of foundations on "real" mathematics and linked to it from foundations.
This is to archive an exchange between Toby and James on the blog here. The hope would be that eventually the list of examples is expanded here.
I’ve wondered for a while whether there is a notion of lax-idempotent 2-adjunction, but for some reason until now I’d never thought to try the obvious route of simply generalizing the conditions defining an idempotent adjunction. Haven’t had time to cross-link it yet.
I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.
I added some references to adjoint triple for the folklore theorem about fully faithful adjoint triples.
I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.
stub on typecase to fix a red link from parametric dependent type theory
starting page on type theories with parametricity, as a separate article from the article on polymorphism.
Anonymouse
Created page. Probably some of the material at transfinite construction of free algebras ought to be moved here, but I’m too lazy to do it right now.
at monadicity theorem in the second formulation of the theorem, item 3, it said
has
I think it must be
has
and have changed it accordingly. But have a look.
added at exact sequence two small lemmas on forming quotients in an exact sequence
Added an article
I have added some minimum content to Stiefel manifold, also a little bit to Grassmannian
reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.
added pointer to
to initial algebra over an endofunctor, higher inductive type and W-type
I have written out a detailed classical point-set proof that the fundamental group of the circle is the integers: here.
I wrote out a proof which uses very little machinery at fundamental theorem of algebra. It is just about at the point where it is not only short and rigorous, but could be understood by an eighteenth-century mathematician. (Nothing important, just fun!)
I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.
added pointer to:
Zhiyuan Wang, Kaden R. A. Hazzard: Particle exchange statistics beyond fermions and bosons, Nature 637 (2025) 314-318 [arXiv:2308.05203, doi:10.1038/s41586-024-08262-7]
Zhiyuan Wang: Parastatistics and a secret communication challenge [arXov:2412.13360]
Created some minimum at Bousfield-Kuhn functor, for the moment just so as to record some references.
added a warning to the top of the page to alert the reader of the issue that some browsers may render the page in a meaningless way (as per discussion here)
But I don’t think the slash notation is used much at all elsewhere on the nLab, so that there is not much to worry about here.
I have created Sullivan model of free loop space with the formula and pointers to the literature.
I got tired of having to fight my way through Kelly’s monster yet again, and created transfinite construction of free algebras. I couldn’t really think of a good name for this page; suggestions are welcome.
stub for jet bundle
I have added to the References at double negation pointer to Andrej’s exposition:
which is really good. I have also added this to double negation transformation, but clearly that entry needs some real references, too.
added to Kan lift the def of absolute Kan lifts, and some more examples.
Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?
Added a “warning” for something that tripped me up: the classifying topos of a classical first-order theory is typically not Boolean, even though the classifying pretopos is Boolean. For a topos to be Boolean is much stronger – as Blass and Scedrov showed, it implies -categoricity.
In order to accompany the nCafe discussion I have started to add some content to the entry Euler characteristic
I am moving the following old query box exchange from orbifold to here.
old query box discussion:
I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.
Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?
Urs Schreiber: please, go ahead. It would be appreciated.
end of old query box discussion
Created W-type.
added pointer to:
I noticed that there was a neglected stub entry universe that failed to link to the fairly detailed (though left in an unpolished state full of forgotten discussions) Grothendieck universe.
I renamed the former to universe > history and made “universe” redirect to “Grothendieck universe”
I have checked with the WaybackMachine for when was the last time it succeeded in taking a snapshot of the “Manifold Atlas” (which is down and has been for a while).
I have added my findings as a little paragraph to the entry:
The site is down and has been for some time. The last successful snapshot made by the WaybackMachine is, for the landing page, from 2024, Dec 26 and, for the “bulletin” page, from 2025, Feb 28. The latest update to the latter dates back to 2017.
So it does look likely that the site is gone for good.
I kept being annoyed about the nature of discussion of the “multiverse” (the one in cosmology, not the one in set theory). Now I thought instead of steadily being annoyed, I should start an Lab entry that does it better. So I did now (or tried to), at multiverse.
Since I was being asked I briefly expanded automorphism infinity-group by adding the internal version and the HoTT syntax.
Mike, what’s the best type theory syntax for the definition of via -image factorization of the name of ?
I rewrote a good bit of the entry sheaf, trying to polish and strengthen the exposition.
The rewritten material is what now constituttes the section “Definition”. This subsumes essentially everything that was there before, except for some scattered remarks which I removed and instad provided hyperlinks for, since they have meanwhile better discussions in other entries.
I left the discussion of sheaves and the general notion of localization untouched (it is now in the section “Sheaves” and localization”). This would now need to be harmonized notationally a bit better. Maybe later.
added pointer to
which provides a wealth of computational details and illustrative graphics.