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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 ℵ0-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 nLab 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 Aut(X) via ∞-image factorization of the name of X?
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.
added link to axiom of relativity (currently a redirect to bridge type, which contains a section on relativity for type universes)
I have added the following paragraph to calculus of constructions, I’d be grateful if experts could briefly give me a sanity check that this is an accurate characterization:
More in detail, the Calculus of (co)Inductive Constructions is
a system of natural deduction with dependent types;
with the natural-deduction rules for dependent product types specified;
and with a rule for how to introduce new such natural-deduction rules for arbitrary (co)inductive types.
and with a type of types (hierarchy).
started Lie algebra cohomology,
(for the moment mainly to record that reference on super Lie algebra cocycles)
I did not change anything, I would not like to do it without Urs’s consent and some opinion. The entry AQFT equates algebraic QFT and axiomatic QFT. In the traditional circle, algebraic quantum field theory meant being based on local nets – local approach of Haag and Araki. This is what the entry now describes. The Weightman axioms are somewhat different, they are based on fields belonging some spaces of distributions, and 30 years ago it was called field axiomatics, unlike the algebraic axiomatics. But these differences are not that important for the main entry on AQFT. What is a bigger drawback is that the third approach to axiomatic QFT if very different and was very strong few decades ago and still has some followers. That is the S-matrix axiomatics which does not believe in physical existence of observables at finite distance, but only in the asymptotic values given by the S-matrix. The first such axiomatics was due Bogoliubov, I think. (Of course he later worked on other approaches, especially on Wightman’s. Both the Wightman’s and Bogoliubov’s formalisms are earlier than the algebraic QFT.)
I would like to say that axiomatic QFT has 3 groups of approaches, and especially to distinguish S-matrix axiomatics from the “algebraic QFT”. Is this disputable ?
wrote a definition and short discussion of covariant derivative in the spirit of oo-Chern-Weil theory
Preprint today by Yau et al., relating p-adic strings to the Riemann zeta function:
added the adelic solenoid as an extension of U(1) by ˆℤ along with references
Dinakar Ramakrishnan, Robert Valenza, Fourier Analysis on Number Fields. Graduate Texts in Mathematics 186, Springer-Verlag, New York, (1999). (doi).
Juan Manuel Burgos, Alberto Verjovsky. Adelic solenoid I: Structure and topology (2016). (arXiv:1603.05676).
a stub entry, to satisfy public demand
a stub entry, to satisfy public demand
Stub context-free grammar.
moved the statement that a profinite group is a group internal to profinite sets from the “Examples”-section to “Definition”-section
I added the definition of a filtered (infinity,1)-category from HTT. Since this is performed in a simplicial model which is supposedly not to be emphasized from the nPov and I felt that the below proposition should center this article I added a sentence indicating this in the ”Idea”.
Created:
\tableofcontents
A quasicategory C is confluent if every cospan Q:Λ20→C the quasicategory IQ/ of cocones under Q is weakly contractible.
If C has pushouts, then C is confluent because the quasicategory of cocones has an initial object.
A quasicategory C is confluent if and only if for every morphism f:A→B, the induced functor f*:CB/→CA/ is a final functor.
If π:T→B is a left fibration and B is confluent, then so is T.
\begin{theorem} (Sattler–Wärn.) A quasicategory C is confluent if and only if C-indexed colimits commute with pullbacks in the quasicategory of ∞-groupoids. \end{theorem}
A quasicategory is filtered if and only if it is confluent and weakly contractible.
Expository account:
I’ve removed this query box from metric space and incorporated its information into the text:
Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched †-categories?
Toby: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched †-categories.
Add example of Eilenberg–Moore objects.
It is being pointed out to me by email that this entry says about the algebra C0(X) of functions vanishing at infinity that:
C0(X) is no longer a Banach space
(due to revision 1 by Todd Trimble, way back in October 2009)
This seems odd, as C0(X) is a standard example of a Banach space, unless something else is meant here.
It seems nothing in the entry depends on this side-remark, so that it may be worth deleting.
I asked in the Examples section
Would Serre classes fit in here? Perhaps that’s one step back.
in a query box, now removed. I (still) don’t know about the answer.
I looked again after a long while at the entry manifold structure of mapping spaces, looking for the statement that for X a compact smooth manifold and Y any smooth manifold, the canonical Frechet structure on C∞(X,Y) coincides with the canonical diffeological structure.
So this statement wasn’t there yet, and hence I have tried to add it, now in Properties – Relation between diffeological and Frechet manifold structure.
To make the layout flow sensibly, I have therefore moved the material that was in the entry previously into its own section, now called Construction of smooth manifold structure on mapping space.
While re-reading the text I found I needed to browse around a good bit to see where some definition is and where some conclusion is. So I thought I’d equip the text more with formal Definition- and Proposition environments and cross-links between them. I started doing so, but maybe I got stuck.
Andrew, when you see this here and have a minute to spare: could you maybe check? I am maybe confused about how the {Pi} and {Qi} are to be read and what the index set of the charts of C∞(M,N) in the end is meant to be. For instance from what you write, what forbids the choice of {Pi} and {Qi} being the singleton consisting just of M and N itself, respectively?
Fixed a hyperlink to Jardine’s lectures. Removed a query box:
+– {: .query} Can any of you size-issue experts help to clarify this?
Mike: I wish. I added some stuff, but I still don’t really understand this business. In particular I don’t really know what is meant by “inessential.” It certainly seems unlikely that you would get equivalent homotopy theories, but it does seem likely that you would get similar behavior no matter where you draw the line. And if all you care about is, say, having a good category of sheaves in which you can embed any particular space or manifold you happen to care about, then that may be good enough. But I don’t really know what the goal is of considering such large sites. =–
edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.
And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?
I think the entry-situation here deserves to be further harmonized.
Created:
\tableofcontents
A variant of a Morse function that yields a contractible space of such functions.
A generalized Morse function f on a smooth manifold is a smooth real-valued function f whose critical points either have a nondegenerate Hessian or a Hessian with a 1-dimensional kernel K such that the third derivative of f along K is nonzero.
The Morse lemma shows that in a neighborhood of such a critical point we can pick a coordinate system in which f has the form
f(x1,…,xn)=x21+⋯+x2k−x2k+1−⋯−x2nor
f(x1,…,xn)=x21+⋯+x2k−1+x3k−x2k+1−⋯−x2n,respectively.
The space of framed generalized Morse functions is contractible. For a proof, see Eliashberg–Mishachev or Kupers.
This property distinguishes framed generalized Morse functions from ordinary Morse functions, whose space is not contractible.
Y. M. Eliashberg, N. M. Mishachev, The space of framed functions is contractible, Essays in Mathematics and its Applications. In Honor of Stephen Smale’s 80th Birthday (2012), 81–109. arXiv, DOI.
Alexander Kupers, Three applications of delooping to h-principles, Geometriae Dedicata 202:1 (2019), 103–151. arXiv, DOI.
there is some confusion on this MO thread about sheafification, with the nLab entry sheafification somehow involved. I had a look at the entry and find that it can do with lots of polishing, but that the statement discussed over there is clearly right. (the misleading answer on MO that seems to claim a problem on the nLab page gets twice as many votes as the good answer by Clark Barwick, which confirms the statement) I have tried to edit it a bit to make things clearer, but don’t have the leisure for that now.
Given the recent success with the polishing of the entry on geometric realization, maybe I should announce that sheafification is going to be submitted for nJournal peer-review soon, so that everybody here will jump on it to brush it up ;-)
I noticed that there was no entry quotient stack, so I quickly started one, just to be able to point to it from elswhere.
I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration
I also edited the "Idea"-section at Grothendieck fibration slightly.
That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.
added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors