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have made explicit the proof that reduced excisive functors are equivalently spectra, here.
started a bare minimum at Adams category (in the process of expanding at stable homotopy category).
I tried adding some material to framed link, but as in other recent edits (Dehn twist, Dehn surgery), my efforts might well make an expert smile indulgently. I do mean to – or someone else could – fix up or polish up the recent edit at Dehn surgery, which I’m not particularly happy with at the moment.
I’ve been adding some things I’ve recently learned to prime ideal theorem.
Added more to Dehn twist which had been in a stubby state.
For no particularly compelling reason (I had a little time before the Superbowl begins), I wrote compact Hausdorff rings are profinite. (You’ll recall that Tom Leinster wrote on this about a year and a half ago at the Café, here.)
I wrote alternating multifunction, which is perhaps too generalized?
I gave Adams’ Stable homotopy and generalised homology its own category:reference-entry, for ease of cross-linking.
stub for Brown-Gitler spectrum (just because I was editing at Samuel Gitler)
I have listed today’s arXiv preprint
at variational calculus despite the title, as it seems that the construction of presymplectic current after Zuckerman’s idea on geometry of variational calculus is very central to the paper.
There was a note at the top of ideal:
This entry discusseds the notion of ideal in fair generality. For an entry closer to the standard notion see at ideal in a monoid.
I've removed this, as it seems exactly backwards. The pages are equally standard, but the most common notion of ideal is that of an ideal in a ring, and that is the first thing discussed at ideal, and in very basic terms; but at ideal in a monoid, this is discussed only via rings' being monoids in , and it's not spelt out.
I will be compiling something that ought to work as lecture notes for a course that introduces stable homotopy theory for people with background in homotopy theory, and aimed at understanding the Adams-Novikov spectral sequence, together with some extra material on the modern picture via descent down to .
Just because as an Lab entry that fits well into the existing growing lecture note series titled “geometry of physics”, I am putting that now into an entry that is titled
(as continuation of the previous geometry of physics – homotopy types) but for the time being there won’t be any physics here, except maybe in the guise of some links on further reading as it gets to the meaning of the stratification of .
For the moment the entry has mainly just the intended skeleton, I will be adjusting that a little more and then start filling it with serious content.
I’m thinking of creating a little page called rigid object about the property of an object of a category having no non-trivial automorphisms. How standard is this terminology? I’ve seen it used in a few places, for example in this paper by Kock et al.. On the other hand, it seems that “rigid object” is also sometimes used to refer to an object of a monoidal category with both left and right duals, as in a rigid monoidal category. Is there any connection between these two usages?
I have started an article well-founded coalgebra, where I’m trying to put together some things I’ve learned while reading Paul Taylor’s work. All comments welcome.
I’ve added references at cohesive topos and Birkhoff’s theorem to Lawvere’s recent paper Birkhoff’s Theorem from a geometric perspective: A simple example, which appears in an obscure Iranian journal.
I have just now two new master students who are going to look into certain geometric aspects of physics. Also a colleague just asked me for suggestions for a course on “geometry and physics”. I kept pointing to Frankel’s book. That’s great as far as it goes, but it misses on a lot of clarifications available meanwhile.
So I thought it’s about time that I start making notes for a modern introductory course on
I put that into the Lab proper, instead of on my personal web. One reason is that otherwise hyperlinking becomes a pain. Another reason is that this should really not be hidden and reserved somewhere but be out there in the open for everyone to join in. Though I do have a certain strategy in mind, which I would like to ask to follow.
You’ll see what I mean when you look at the entry. It’s so far just a first sketch of a section outline with some keywords and notes to indicate what is eventually to go there. That’s how far I got tonight. (And I really need to sleep now to be ready for my homological algebra course tomorrow…) But I guess the idea and the intended structure is already visible. Will be expanded and edited in the course of the next weeks.
I have copied the nice implication flow chart from Adams’ original paper into the entry, here
mentioning it here because I don’t want to sidetrack the other thread, but I went ahead and added some context around that comment from John Baez quoted in virtual knot theory and tried to make it more qualified. Feel free to further adjust the wording.
I am giving Modern foundations for stable homotopy theory a category:reference entry.
First thing I did was to brush-up the list of references at symmetric smash product of spectra. Then I copied over the nicely to-the-point History-paragraph to a new section stable homotopy theory – history.
I added the bare statement of the list of conditions to Artin-Lurie representability theorem, and then added the remark highlighting that the clause on “infinitesimal cohesion” implies that the Lie differentiation of any DM -stack at any point is a formal moduli problem, hence equivalently an -algebra. Made the corresponding remark more explicit also at cohesive (∞,1)-presheaf on E-∞ rings.
New stub stochastic quantization.
Added appropriate axioms for the various definitions of affine space, along with another definition in terms of a single quaternary operation.
I created classifying (infinity,1)-topos. It links to the special case discussed at structured spaces. The example section is in classifying topos, too.
for completeness, I gave formal neighbourhood of the diagonal its own little entry.
Stub for localic completion. I wonder to what extent this can be generalized beyond metric spaces; for uniform spaces or Cauchy spaces we don’t have a nice collection of canonical basis elements like the open balls.
I started an article, surreal number. I’ve run out of energy to put in all the links that should be in there.
I want to get at Conway games at some point, as they are more basic than Conway numbers, and fit well within an nPOV. Conway discovered numbers after games, and it seems only right to establish that priority also within the Lab. In particular, one should mention Joyal’s description of the category of games, and what this has to do with the ordering on numbers.
I have edited the second point under examples on the cogroup page. I replaced what I believe to be an erroneous with , and have included a reference for the claim that there are cogroups in which are not suspensions.
am starting an entry tangent complex. For the moment its biggest achievement is to give a pointer to section 8 in Hinich’s invaluable Homological algebra of homotopy algebras .
I have created an entry-for-inclusion Goodwillie calculus - contents, and have included it as a “floating table of contents” into the relevant entries.
I gave Tate spectrum a bare minimum of content.
created analytic (∞,1)-functor.
There is clearly some deep relation here to the Blakers-Massey theorem. But I am not sure yet what the full picture is.
Created a stub entry for norm map, for the moment just so as to make cross-links work.
stub for moduli space of connections, started to collect some references
I have added some bare minimum content to EHP spectral sequence.
I have added some minimum (or not even that) to p-completion. In the process I also created analytic completion and gave fracture theorem an Idea-section.
(None of this is meant to be in the state in which it is, that’s just how far I got in little available time…)
in order toput things in perspective, I created a table
and included it into relevant entries (under “Related concepts”)
I am back to working on geometry of physics. I’ll be out-sourcing new paragraphs there to their own Lab entries as much as possible (because the length of the page makes saving and hence previewing it take many minutes, so I need to work in smaller sub-entries and then copy-and-paste).
In this context I now started an entry prequantum field theory. To be further expanded.
This comes with a table of related concepts extended prequantum field theory - table:
extended prequantum field theory
| transgression to dimension | |
|---|---|
| extended Lagrangian, universal characteristic map | |
| (off-shell) prequantum (n-k)-bundle | |
| (off-shell) prequantum circle bundle | |
| action functional = prequantum 0-bundle |
Continuing from a very minor edit on localic topos, I've created articles on first-order hyperdoctrines and triposes; both need fleshing out, but the latter in particular I've only just barely started. I intend to add to it a more explicit description of the construction of a topos from a tripos, and discussion of some specific examples (those given by complete Heyting algebras and by realizability relative to a partial combinatory algebra). (Also, the definition has only been given for a special case at the moment).
New entry groupoid quantale so far covering just the construction for the discrete case. But the Resende’s paper cited therein goes of course much beyond.
I gave André Joyal’s lectures in Paris last week their own category:reference page on the Lab, in order to be able to link to them conveniently (from entries such as topos theory and (infinity,1)-topos theory):
I’ve started relational beta-module. It would be lovely if somebody who really grasps it could fill in the abstract definition and maybe check (or even show how to derive) the concrete one, which I extracted from this blog post by Todd Trimble. (Hey, maybe Todd could check it!)
This started when I realised that being infinitely close is a uniform (not topological) property in nonstandard analysis, which is hinted at by the very bottom of the page (as it is now).
Does someone know offhand the relationship between the stabilization hypothesis “for -categories” attributed to Joyal and Lurie at stabilization hypothesis and the version that appears in arXiv:1312.3178? It would be nice to add a reference to the latter to the page stabilization hypothesis but I’m not sure how to relate it to what’s already there.
At present the entry EGA is not only about EGA but includes sections on FGA and SGA. Should it be renamed and a new page with that title be created which can do what is said will be done there (e.g. list of chapters etc.).
I have finally added a little bit of substance to Polyakov action (with a little spill-over at Nambu-Goto action).
This is not polished yet, I need to run now and come back to it later.
stub at locally compact locale
Someone unhelpfully started Gauss lemma without any content. It was required by good open cover.
According to Wikipedia there needs to be disambiguation.
I am making lots of little edits on F-theory related entries, mostly adding references and pointers to them with brief comments. Hence nothing that deserves much announcement here, but just in case you are watchign the logs and are wondering, I’ll announce some of it anyway, trivial as it may be.
So at supersymmetry and Calabi-Yau manifolds (which exists since long ago but was maybe never announced in the first place, so now it is) I have included a table-for-inclusion titled “N=1 susy compactifications – table”, and also included it then at M-theory on G2-manifolds and at F-theory on CY4-manifolds.
At stable model category I have tried to brush up the section (which is now titled) Properties – As A-infinity algebroid module categories.
Adeel is of course invited to expand further…
added to noncommutative motive a brief version of the Definition due to Blumberg-Gepner-Tabuada.
Also added (with brief comments) their references and the dg-category theoretic precursors by Denis-Charles Cisinski and Tabuada.
(Deserves to be expanded further, certainly, just a quick note so far.)
I have (finally) added some pointers to the result of Freed-Hopkins 13 to relevant Lab entries.
Mostly at Weil algebra – characterization in the smooth infinity-topos
also at invariant polynomials – As differential forms on the moduli stack of connections
pointing out that this adds further rationalization to the construction of connections on principal infinity-bundles – via Lie integration.
In making these edits, I have created and then used a little table-for-inclusion
Presently this displays as follows:
Chevalley-Eilenberg algebra CE Weil algebra W invariant polynomials inv
differential forms on moduli stack of principal connections (Freed-Hopkins 13):
I have extracted one of the key statements from
to an entry algebraic K-theory of smooth manifolds.
Someone created a page ’www.emis.de/journals/AM/09-4/roger.ps.gz’. I thought maybe it had come from putting URL before title, but this isn’t so for the three pages which refer to that paper. I’ve changed those three links to the pdf version.
I got tired of not having linear combination, so now we have it.
I created stub for Schwarzian derivative just to record several references.
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?