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I have created a page list of mathematics software with links to all the nLab pages I could find about software packages, and put all of those pages in category: software.
An anonymous coward put something blank (or possibly some spam that somebody else blanked within half an hour) at Hausdorff dimension, so I put in a stub.
added section on Russell’s relation to mysticism based on his essay Mysticism and Logic and the biography of Ray Monk.
Created a little entry Vect(X) (to go along with Vect) and used the occasion to give distributive monoidal category the Examples-section that it was missing and similarly touched the Examples-section at rig category.
added pointer to today’s
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)?
Significant improvements and expansion in determinant line bundle and new related stub analytic torsion.
Changed ’∞-compact’ to ’sigma-compact’.
This page is a bit weird, because apparently in 2018 or so I had an idea to generalise the notion in the literature, but didn’t explain it enough for me to reconstruct my idea. I think that the real definition should go here, but I will come back to this soon. Possibly some of the classical stuff works here; I’m thinking the source-fibre-wise Haar measure might exist, but maybe not. But not everything will work, since the space of objects not being locally compact is pretty fatal for getting an overall measure.
I have added the adjoint modality of Even⊣Odd on (ℤ,≤).
This example is from adjoint modality (here). But it was actually a little wrong there. I have fixed it and expanded there and then copied over to here.
finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).
I’ll be preparing here notes for my lectures Categories and Toposes (schreiber), later this month.
Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.
Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.
I would try to say what the idea is, but have to dash.
a bare subsection with a list of references, to be !include-ed at super Riemann surface and at moduli space of super Riemann surfaces, for ease of synchronization
created a stub for super Riemann surface, just to record Witten’s latest
I have added a little bit to supermanifold, mainly the definition as manifolds over superpoints, the statement of the equivalence to the locally-ringed-space definition and references.
At Fréchet space I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of ℝ∞=lim⟵nℝn. And I touched the description of this example in the main text, now here.
Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category Set. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group G on a set X, and looks what happens in the vector space of functions into a field K. As we know, for a group element g the definition is, (gf)(x)=f(g−1x), for f:X→K is the way to induce a representation on the function space KX. The latter representation is called the permutation representation in the standard representation theory books like in
I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.
Edit: new (related) entries for Claudio Procesi and Arun Ram.
Added:
There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces.
Later Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. However, many authors continue to use the original definition due to Banach.
The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion.
The nLab uses “F-space” to refer to the non-locally convex notion and “Fréchet space” to refer to the locally convex notion.
I added a Definition section to Burnside ring (and made Burnside rig redirect to it).
stub for quantum computation
added at adjoint functor
more details in the section In terms of universal arrows;
a bit in the section Examples
Added to Dedekind cut a short remark on the ¬¬-stability of membership in the lower resp. the upper set of a Dedekind cut.
I have added some accompanying text to the list of links at monad (disambiguation).
One question: in the entry Gottfried Leibniz it is claimed that the term “monad” for a functor on a category with monoid structure also follows Leibniz’s notion of monads. Is this really so? What’s a reference for this claim?
I am asking because I don’t see how the notion of monoid in the endomorphisms of a category would be related to what Leibniz was talking about. What’s the idea, if there is one?
added reference to derived category
added pointer to:
Julian Schwinger: Quantum Kinematics and Dynamics, CRC Press (1969, 1991) [ISBN:9780738203034, pdf]
Julian Schwinger (ed.: Berthold-Georg Englert): Quantum Mechanics – Symbolism of Atomic Measurements, Springer (2001) [doi:10.1007/978-3-662-04589-3]
with a link to arguments that Schwinger secretly (re-)invented groupoid algebra, in these books.
a bare list of references on arguments
(by Connes) that Heisenberg’s original derivation of “matrix mechanics” and
more generally (by Ibort et al.) that Schwinger’s less known “algebra of selective measurements”
are both best understood, in modern language, as groupoid convolution algebras,
to be !include-ed into relevant entries (such as quantum observables and groupoid algebra), for ease of synchronizing
a stub entry, for the moment just to make a link work that has long been requested at Handbook of Quantum Gravity
for the equivariant+twisted version I added further pointer to
El-kaïoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014), no.5 (arXiv:1202.2057)
Daniel Freed, Gregory Moore, Section 7 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)
Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134, spire:1601772)