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polished and expanded adjoint (infinity,1)-functor
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 ?
Added
For a treatment in homotopy type theory see
- Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide, Finite Sets in Homotopy Type Theory, (pdf)
created computational trinitarianism, combining a pointer to an exposition by Bob Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.
In discrete fibration I added a new section on the Street’s definition of a discrete fibration from to , that is the version for spans of internal categories. I do not really understand this added definition, so if somebody has comments or further clarifications…
Added
Sounds interesting
We would like to mention that a structure similar to categorical symmetry was found previously in AdS/CFT correspondence, [63–65] where a global symmetry at the high-energy boundary is related to a gauge theory of group in the low-energy bulk. In this paper, we stress that the categorical symmetry encoded by the bulk -gauge theory not only contains the symmetry at the boundary, it also contains a dual algebraic higher symmetry at the boundary.
I created the article multisimplicial set.
It seems that I triggered some bug in Instiki that prevents it from rendering displayed formulas correctly.
The article compiles just fine in TeX, so this is clearly a bug.
In fact, Instiki seems to insert some new text:
definedasthechunk29161400wikichunklinkchunkofthetautologicalfunctor
All this “chunk” and “wikichunklink” stuff is clearly a bug.
started something stubby at Liouville theory, for the moment just so as to record some references and provide for a minimum of cross-links (e.g. with Chern-Simons gravity).
(also created a stub for quantum Teichmüller theory in the course of this, but nothing there yet except a pointer to reviews)
recorded two references to physics application at exotic smooth structure
(no time for anything else)
started universal exceptionalism
related discussion is taking place on g+ here
added this to the list of references:
Speculative remarks on the possible role of maps from spacetime to the 4-sphere in some kind of quantum gravity via spectral geometry are in
reviewed in
Alain Connes, section 4 of Geometry and the Quantum, Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham, 2018. 159-196 (arXiv:1703.02470)
Alain Connes, from 58:00 to 1:25:00 in Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG, talk at IHES, 2017 (video recording)
Strangely, we don’t seem to have an nForum discussion for probability theory.
I added a reference there to
It replaces the category of measurable spaces, which isn’t cartesian closed, with the category of quasi-Borel spaces, which is. As they point out in section IX, what they’re doing is working with concrete sheaves on an established category of spaces, rather like the move to diffeological spaces.
[Given the interest in topology around these parts at the moment, we hear of ’C-spaces’ as generalized topological spaces arising from a similar sheaf construction in C. Xu and M. Escardo, “A constructive model of uniform continuity,” in Proc. TLCA, 2013.]
For no particular reason, I have added another illustrating graphics to the entry, taken from Fig 1.3 in Apostol 1973.
I noticed that the entry analysis is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at References – General
At interactions of images and pre-images with unions and intersections I have added pointer to Lawvere 69 and there at Adjointness in Foundations I added a bit more text and cross-references.
Started an article on monoidal monad. An earlier redirect had sent it over to Hopf monad which is something that Zoran was working on, but I think it deserves an article to itself, with discussion of the relation to commutative monads, etc. (which I have started).
The term “tensor network”, while essentially just a synonym for “string diagram”, has in recent years become widely used and now fully established in quantum physics, especially in its use for discussion of holographic entanglement entropy. It needs a page of its own, if only to point to string diagram while also listing the relevant physics references
added pointer to
Since it was mentioned by Urs on g+, I thought I’d start mysterious duality. Maybe not a great name when someone discovers how it works (as someone claims to have done here).
Stub for Delta-generated space.
addes some sentences concerning the Quillen model structure to model structure on topological spaces
David C. has kindly created a list of references at Geometry of Interaction. I have now started there an Idea section:
What has been called Geometry of Interaction (Girard 89) is a kind of semantics for linear logic/linear type theory that is however different in method from the usual categorical semantics in monoidal categories. Instead of interpreting a proof of linear implication as a morphism between objects and in a monoidal category as in categorical semantics, the Geometry of Interaction interprets it as an endomorphism on the object . This has been named operational semantics to contrast with the traditional denotational semantics.
That also the “operational semantics” of GoI has an interpretation in category theory, though, namely in traced monoidal categories was first suggested in (Joyal-Street-Verity 96) and then developed out in (Haghverdi 00, Abramsky-Haghverdi-Scott 02,Haghverdi-Scott 05).
I have changed the title of this article, as well as references to the object within it. Use of the term “Hawaiian Earring” is objected to by Hawaiian mathematicians. Please see these two threads, one by native Hawaiian and math PhD Dr. Marissa Loving, and the other by an expert on the Hawaiian Earring, Dr. Jeremy Brazas.
https://twitter.com/MarissaKawehi/status/1406244897611522049
https://twitter.com/jtbrazas/status/1406652385263501319
I have retitled the article “Shrinking wedge of circles”, which is the name used for this space in Hatcher’s “Algebraic Topology”. I have a retained a note in the body of the article that the space is sometimes referred to as the “Hawaiian earring space”.
This small change in name helps to make mathematics a more inclusive and just field, especially in consideration of the historical marginalization and exclusion of indigenous mathematicians. By taking this action, the nLab site can help to spread a change in language more widely, including on other math reference sites.
I hope that this change is readily accepted and approved by the nLab community. Thank you!
Justin Lanier
added at cobordism hypothesis a pointer to
where the case for -categories is spelled out and proven in detail.
noticed that we didn’t have complex Lie group, so I put in a bare minimum. Also a stub for complex Lie algebra. Just so that it’s there.
added a table with some homotopy groups in the unstable range to orthogonal group – Homotopy groups
Added a diagram at commutative algebraic theory using the totally awesome SVG Editor.
it now does itex!
This was a ridiculously simple diagram to do.
added pointer to
In the article connection on a cubical set the definition of a connection uses two types of maps, denoted Γ^+_i and Γ^-_i. Roughly, the former corresponds to a map of cubes that takes the minimum of some coordinates, whereas the latter takes the maximum of some coordinates.
However, in the book by Brown-Higgins-Sivera in Definition 13.1.3, page 446, only the maps Γ^-_i are used. There they are denoted simply by Γ_i. The paper by Maltsiniotis about the strict test category of cubes with connection also uses the same definition.
Which definition is correct? What is the reference for nLab’s definition and why does it deviate from the definition of Brown-Higgins-Sivera?
In the past we had some discussion here about why simplicial methods find so much more attention than cubical methods in higher category theory. The reply (as far as I am concerned at least) has been: because the homotopy theory = weak oo-groupoid theory happens to be well developed for simplicial sets and not so well developed for cubical sets. Historically this apparently goes back to the disappointment that the standard cubical geometric realization to Top does not behave as nicely as the one on simplicial sets does.
Still, it should be useful to have as much cubical homotopy theory around as possible. Many structures are more naturally cubical than simplicial.
So as soon as the Lab comes up again (we are working on it...) I want to create a page model structure on cubical sets and record for instance this reference here:
Jardine, Cubical homotopy theory: a beginning
Made some further tweaks at cubical set. Hopefully the definitions of the boundary functor, and of a horn, are correct now.
Am continuing to work on homotopy groups of a cubical Kan complex.
Added another reference.
I was chatting with Robin Cockett yesterday at SYCO1. In a talk Robin claims to be after
The algebraic/categorical foundations for differential calculus and differential geometry.
It would be good to see how this approach compares with differential cohesive HoTT.
This appears to be called Stigler’s law of eponymy on Wikipedia. Is there good reason to have a separate term?
I am splitting off Gelfand duality from Gelfand spectrum. Want to state the actual equivalence theorem here. But just a moment…
tried to improve the entry coproduct a little
I have touched the formatting at direct sum and then expanded a little:
Added a paragraph to the Idea-section such that something familiar is mentioned right at the beginning;
Expanded on the example of direct sums in by drawing the cocone diagrams and explicitly mentioning the universal property.
Mentioned the relation to formal linear combinations.
Mentioned the examples of direct sums of modules.
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.
New entry linearly compact module with many redirects, just to record some references.
New book entry Abelian categories with applications to rings and modules, to record its contents.