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Someone with pseudonym Z has posted a question at the bottom of a query box at hyperstructure. The question is:
Z: I’d like to know more about composition of bonds as described on p.8 of “Higher Order Architecture of Collections of Objects” (Nils Baas). Can someone please clarify the rules on this page?
noticed that the entry was missing, so I created a stub for integer
somebody asked me for the proof of the claim at canonical topology that for a Grothendieck topos H we have H≃Shcan(H).
I have added to the entry pointers to the proof in Johnstone’s book, and to related discussion for ∞-toposes. Myself I don’t have more time right now, but maybe somebody feels inspired to write out some details in the nLab entry itself?
Someone anonymous has raised the question of subdivision at cellular approximation theorem. I do not have a source here in which I can check this. Can anyone else check up?
Created:
Nikolai Mishachev (Николай Мишачёв) is a Russian mathematician working on h-principle and related topics.
!redirects N. M. Mishachev !redirects N. Mishachev !redirects Н. М. Мишачёв !redirects Н. Мишачёв !redirects Николай Мишачёв
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.
added below the very first definition at kernel a remark that spells out the universal property more explicitly. Also added mentioning of some basic examples.
category: people page for preprint
Anonymouse
category: people page for preprint
Anonymouse
category: people page for preprint
Anonymouse
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]
category: people page for reference
Anonymouse
category: people page for reference
Anonymouse
I have tried to polish polynomial a little.
category: people page for reference
Anonymouse
Danny Stevenson was so kind and completed spelling out the proof of the pasting law for ∞-pullbacks here at (infinity,1)-pullback.
starting page on (infinity,1)-equalizers with references
Jacob Lurie, section 4.4.3 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)
Denis-Charles Cisinski, Bastiaan Cnossen, Kim Nguyen, Tashi Walde, section 5.6 of: Formalization of Higher Categories, work in progress (pdf)
that make it clear that they are talking about coequalizers in (infinity,1)-categories and not homotopy coequalizers in model categories
Anonymouse
replacing nonfunctional link to website with the reference
Anonymouse
I added two new important references on global analytic geometry, also due to Poineau. He shows there that the sheaf of analytic functions is coherent. This is an interesting fundamental result.
created website-link page Denis-Charles Cisinski
category: people page for reference
Anonymouse
I fixed the first sentence at doctrine. It used to say
A doctrine, as the word was originally used by Jon Beck, is a categorification of a “theory”.
I have changed it to
The concept of doctrine, as the word was originally used by Jon Beck, is a categorification of the concept of “theory”.
If you see what I mean.
Then I added to the References-section this:
The word “doctrine” itself is entirely due to Jon Beck and signifies something which is like a theory, except appropriate to be interpreted in the category of categories, rather than, for example, in the category of sets; of course, an important example of a doctrine is a 2-monad, and among 2-monads there are key examples whose category of “algebras” is actually a category of theories in the set-interpretable sense. Among such “theories of theories”, there is a special kind whose study I proposed in that paper. This kind has come to be known as “Kock-Zoeberlein” doctrine in honor of those who first worked out some of the basic properties and ramifications, but the recognition of its probable importance had emerged from those discussions with Jon.
Updating Frédéric Paugam reference to
since the previous reference was just a website link that no longer worked.
Anonymouse
category: people page for reference
Anonymouse