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I started finite ∞-group, and added that same reference to Sylow p-subgroup.
following a suggestion by Zoran, I have created a stub (nothing more) for Kuiper’s theorem
At first Zoran's reply to my query at structured (infinity,1)-topos sounded as though he were saying "being idempotent-complete" were a structure on an (oo,1)-category rather than just a property of it. That had me worried for a while. It looks, though, like what he meant is that "being idempotent" is structure rather than a property, and that makes perfect sense. So I created idempotent complete (infinity,1)-category.
Stack entry says: "The notion of stack is the one-step vertical categorification of a sheaf." In Grothendieck's main works, like pursuing stacks and in the following works of French schools, stack is any-times categorification of a sheaf, and the one-step case is called more specifically 1-stack. We can talk thus about stack in narrow sense or 1-stacks and stacks in wider sense as n-stacks for all n. Topos literature mainly means that the stack is the same as internal 1-stack.
I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.
created univalence axiom
After sitting on this for days and hardly doing a thing, I added some applications to distribution and added a bit to the section on synthetic differential geometry. While I was dawdling, Andrew Stacey stepped in and added to some parts that needed expert attention -- thanks, Andrew.
I have started to add some of the basic definitions and facts to Schwartz space, tempered distribution and Fourier transform of distributions.
Notice that we had an entry titled “Schwartz space” already since May 2013 (rev 1 by Andrew Stacey) which considered not spaces of smooth functions with rapidly decreasing derivatives, but locally convex TVSs “with the property that whenever is an absolutely convex neighbourhood of then it contains another, say , such that maps to a precompact set in the normed vector space .”
I had not been aware of this use of “Schwartz space” before, and Andrew gave neither reference nor discussion of the evident question, whether “the” Schwartz space is “a” Schwartz space. In June 2015 somebody saw our entry and shared his confusion about this point on Maths.SE here, with no reply so far.
I see that this other use of “Schwartz space” appears in Terzioglu 69 (web) where it is attributed to Grothendieck.
Created extranatural transformation by moving the relevant information from dinatural transformation and adding the definition. Disagreements are welcome, but I feel that since dinaturals that aren't extranatural are so rare and harder to deal with and understand, extranaturals merit their own page.
New entry distribution of subspaces and a disambiguating remark at distribution.
I have been adding some more (historical) references to the entry quantum electrodynamics (also at quantum field theory, S-matrix and causal propagator)
have been writing some Idea-section at causal perturbation theory
(currently this has much overlap with locally covariant perturbative quantum field theory, eventially the latter will contain more stuff)
Stephan Alexander Spahn has created descent object, with some definitions from Street’s Categorical and combinatorial aspects of descent theory.
If I get the opportunity this weekend I’ll add details from Street’s Correction to ’Fibrations in bicategories’ and Lack’s Codescent objects and coherence. Anyone know of any other references?
Looking at Street’s paper again, what he describes as the ’’ case of codescent objects looks to be just the notion of a coequalizer. I would have expected reflexive coequalizers, though, because the higher- case uses -truncated simplicial objects. Is there a reason for this?
[New thread because, although it existed since 2012, pasting scheme appears not to have had a LatestChanges thread]
Started to expand pasting schemes. Intend to do more on this soon, in an integrated fashion with digraph and planar graph.
PLEASE note: ACCIDENTALLY a page pasting schemes was created too, as a result of some arcane issues with pluralized names of pages-still-empty. Please delete pasting schemes.
created pullback of a distribution, just for completeness
have created extension of distributions with the statement of the characterization of the space of point-extensions of distributions of finite degree of divergence: here
This space is what gets identified as the space of renormalization freedom (counter-terms) in the formalization of perturbative renormalization of QFT in the approach of “causal perturbation theory”. Accordingly, the references for the theorem, as far as I am aware, are from the mathematical physics literature, going back to Epstein-Glaser 73. But the statement as such stands independently of its application to QFT, is fairly elementary and clearly of interest in itself. If anyone knows reference in the pure mathematics literature (earlier or independent or with more general statements that easily reduce to this one), please let me know.
brief note on Whitney extension theorem
added an Idea-section to Mackey functor (which used to be just a list of references). Also added more references.
created a stub for almost Kähler geometric quantization, for the moment just so as to collect some pointers.
created a stub for pseudoholomorphic vector bundle, for the moment just to record references
started a stub for projective space
added to S-matrix a useful historical comment by Ron Maimon (see there for citation)
I gave product of distributions its own entry. For the moment it just points to the definition in Hörmander’s book.
This should eventually supercede the section “Multiplication of distributions” at distributions, which I find suboptimal: that section starts very vaguely referring to physics as if the issue only appears there, and it keeps being very vague, with its three sub-subsections being little more than a pointer to one reference by Colombeau.
I suggest to
remove that whole subsection at distribution and leave just a pointer to product of distributions
move the mentioning of Colombeau’s reference to product of distributions and say how it relates to Hörmander’s definition
remove all vague mentioning of application in physics and instead add a pointer to Wick algebra and microcausal functional, which I will create shortly.
created a minimum at tensor product of distributions
I have given interaction picture genuine content (the entry used to be effectively empty):
gave it one section “In quantum mechanics” with the standard kind of material going from interacting Hamiltonians to the definition of the S-matrix, and then a section “In quantum field theory” with an outline of which steps in the previous discussion require special technical care and how.
In the process I expanded the entry Dyson formula. (In the end I effectively rewrote it, but now with a little broader perspective and more pointers).
at Beck-Chevalley condition I have added an Examples-section Pullbacks of opfibrations with statement and proof that the diagram of presheaves induced by a pullback of a small obfibration satisfies Beck-Chevalley.
at distribution there used to be a mentioning of “Colombeau algebras”. I have now removed that paragraph there, and have given it its stand-alone stub entry Colombeau algebra, expanding it slightly.
An expert might want to check. I haven’t actually looked into Colombeau algebras beyond a scanning of a review, and presently I don’t plan to delve into the topic. In fact their idea looks misguided to me.
All I mean to do here is to clean up the structure of the entry distribution (see also my comments in the thread on products of distibutions, here) while preserving what others had written before.
the page action is also a mess. I have added a pointer to the somewhat more comprehensive module and am hereby moving the following discussion box from there to here:
[ begin forwarded discussion ]
+–{.query} I am wondering if we will need the notion of action which works in categories with product, i.e. and so on. There is also an action of one Lie algebra on another (for instance in some definitions of crossed module of Lie algebra, where is replaced by the Lie algebra of derivations. (a similar situation would seem to exist in various other categories where action is needed in a slightly wider context. I think most would be covered by an enriched setting but I am not sure.) Thoughts please.Tim
Yes, I think certainly all those types of action should eventually be described somewhere, possibly on this page. -Mike
Tim: I have added some of this above. There should be mention of actions of a monoid in a monoidal category on other objects, perhaps.
Mac Lane, VII.4, only requires a monoidal category to define actions. – Uday =–
[ end forwarded discussion ]
created adiabatic switching (in quantum field theory) with a survey of the idea and references.
created point-supported distribution