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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
started G-CW complex.
I added some material to Peano arithmetic and Robinson arithmetic. At the latter, I replaced the word “fragment” (which sounds off to my ears – actually Wikipedia talks about thisterm a little) with “weakening”.
Still some links to be inserted.
This is a short article with a definition of the category of filters, now called the category of filters.
I apologise for vandalising by mistake the article on filters, and thanks to Richard Williamson for fixing this. I tried to “edit a current page .. in context on a relevant page” misunderstanding the intructions from HowTo:
How to start a new page
You do this in two steps, the first of which may have already been done:
Create a preliminary link (represented by a question mark) by editing > a current page and putting the name of the new page in double square brackets
Anonymous
at subobject classifier I have cleaned up the statement of the definition and then indicated the proof that in locally small categories subobject classifiers precisely represent the subobject-presheaf.
added pointer to
to initial algebra over an endofunctor, higher inductive type and W-type
How would people feel about renaming distributor to profunctor? I seem to recall that when this came up on the Cafe, I was the main proponent of the former over the latter, and I've since changed my mind.
I made a little addition to opposite category, pointing out some amusing nuances regarding the opposite of a V-enriched category when V is merely braided. This remark could surely be clarified, but I think you’ll get the idea.
(In case you’re wondering why I did this, it’s because I needed a reference for “opposite category” in a blog entry I’m writing.)
added English translation of this bit
PN§260 Der Raum ist in sich selbst der Widerspruch des gleichgültigen Auseinanderseins und der unterschiedlosen Kontinuität, die reine Negativität seiner selbst und das Übergehen zunächst in die Zeit. Ebenso ist die Zeit, da deren in Eins zusammengehaltene entgegengesetzte Momente sich unmittelbar aufheben, das unmittelbare Zusammenfallen in die Indifferenz, in das ununterschiedene Außereinander oder den Raum.
Space is in itself the contradiction of the indifferent being-apart and of the difference-less continuity, the pure negativity of itself and the transformation, first of all, to time. In the same manner time – since its opposite moments, held together in unity, immeditely sublate themselves – is the undifferentiated being-apart or: space.
And polished a little around and following this bit.
added the full definition to factorization algebra
In the category:people-entry “William Lawvere” I have created a subsection “Motivation from foundations of physics” where I want to collect pointers to where and how Lawvere was/is motivated from finding foundations for (classical continuum) physics.
Explicit evidence for this that I am aware of includes notably the texts Toposes of laws of motion and the introduction to the book Categories in Continuum Physics.
The Wikipedia entry has this about motivation from physics:
Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell. He learned of category theory [...] found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. [...] meeting on “Categories in Continuum Physics” in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the synthetic differential geometry which had evolved from the spatial part of Lawvere’s categorical dynamics program). Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.
Question: Can anyone point me to more on this early phase of the story (graduate student is supposed to start to look into continuum mechanics, starts to wonder “What is a vector field, really?, what a differential equation?” and ends up revolutionizing the foundations of differential calculus)?
I just aadded a sentence about Yang-Mills theory to gauge group, but there are some aspects of that article I feel we might want to discuss:
I don’t think that the statement “gauge groups encoded redundancies” of the mathematical description of the physics is correct. One hears this every now and then, and I suppose the idea is the observation that physical observables have to be in the trivial representation of the gauge group, but there is more to the gauge group than that.
Notably Yang-Mills theory is a theory of connections on G-principal bundles. No mathematician would ever say that the group G in a G-principal bundle just encodes a redundancy of our descriptins of that bundle. And the reason is because it is true only locally: the thing is that BG={*g∈G→*} has a single object and hence is connected , but it has higher homotopy groups, and that’s where all the important information encoded by the gauge group sits.
So I would say that instead of being a redundancy of the description, instead the gauge group of Yang-Mills theory enocedes precisely the homotopy type of its moduli space. This is rather important.
A different matter are global gauge symmetries such as those that the DHR-theory deals with.
added pointer to
Peter Arndt, section 3.2 of Abstract motivic homotopy theory, thesis 2017 (web, pdf, ArndtAbstractMotivic.pdf:file)
exposition: lecture at Geometry in Modal HoTT, 2019 (recording I, recording II)
following discussion here, this is a page with a bare section, to be !include-ed into relevant entries (at wallpaper group, where it serves as illustration and at G-CW complex where it serves as examples)
It’s not hard or surprising to produce these equivariant cell decompositions of the 2-torus (though it’s generally instructive and it is a prerequisite for computing their sufficiently general equivariant cohomology) but I have never seen it made explicit anywhere else. If anyone has a reference, please drop me a note.
some minimum (on the notion in solid state physics, the other now renamed to crystal (algebraic geometry))
added pointer to:
Eugene P. Wigner Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Springer (1931) [doi:10.1007/978-3-663-02555-9, pdf]
Eugene P. Wigner, Group theory: And its application to the quantum mechanics of atomic spectra, 5, Academic Press (1959) [doi:978-0-12-750550-3]
Added a doi for the Ehrig reference, and copied over the full details of the Lack-Sobocinski reference from adhesive categories.
Added a (sketchy) pointer to
This page just had a couple of references, so I’ve added the idea and more references.
I came to this subject via Lurie’s MO question. Isn’t it a shame that such a highly regarded mathematician reaching out to the categorical logic community doesn’t receive an answer from them?
created some minimum at Cardy condition.
Back then some kind soul provided these cobordism pictures at Frobenius algebra. Is that somebody still around and might easily provide also the picture for the Cardy condition?
Requested clarification in the currently false/confusing entry universal+differential+envelope (sorry I couldn't do more: this will need someone more qualified than me). -- Benoit Jubin
just a minimum for the moment, in order to record the definition in:
Start to a page for the PhysLean project. This relates to the pages on Lean, and projects like the Xena project. To develop it further, an inclusion of exmples of results formalized.
The title appears to have a space in it, and I’m not sure how to deal with this. I can investigate.
Joseph Tooby-Smith
Wrote an article Eudoxus real number, a concept due to Schanuel.
prompted by a question by email, I have expanded at homotopy pullback the section on Concrete constructions by listing and discussing the precise conditions under which ordinary pullbacks are homotopy pullbacks.
Most of this information is scattered around elswehere on the nLab (such as at homotopy limit and right proper model category) and I had wrongly believed that it was already collected here. But it wasn’t.
added references to Lean
I wrote something at meaning explanation, but I didn’t add any links to it yet because I’m hoping to get some feedback from type theorists as to its correctness (or lack thereof).
(…)
added to (infinity,2)-category a section models for the (oo,1)-category of all (oo,2)-categories
I also added (infinity,2)-category and Theta-space to the floating TOC
started bracket type, just for completeness, but don’t really have time for it