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in another entry I want to be able to point to context extension, so I created a brief entry
collected some references at model structure for n-excisive functors and added cross links.
I had not been aware before that Lydakis also got a symmetric monoidal smash on the model structure for excisive functors.
gave minimal Kan fibration some actual content. Definition, list of basic properties, brief indication of their proofs, and some comments on the broader story in the Idea-section.
Added some comments to combinatorial spectrum about their relationship to other spectra and to modern "brave new algebra" (moved up out of the query-box discussion).
I have cleaned up and then expanded a little the entry simplicial homotopy group.
(There used to be various suggestions in the entry, in the main text and in the query boxes, that the simplicial homotopy groups should be organized into an n-groupoid structure. I had complained about that in the query boxes long time back, and they had been sitting there since, now I have removed them all to make the entry not look as awkward anymore. )
I changed the name of discrete space to discrete object such that it is now consistent with codiscrete object.
I wrote Cantor-Schroeder-Bernstein theorem.
created an entry multiplicative spectral sequence with essentially the notes that Sebastian Goette had recently put on MO (here and here)
added to Quillen adjunction the statement how the SSet-enriched version presents adjoint (infinity,1)-functors.
Also indicated how one shows that a left Quillen functor prserves weak equivalences between cofibrant objects, and dually
I created prestable ∞-category.
After having a window open to do edits at shelf for almost the past week, I’ve decided just to hit submit. Would like to get back to this, especially to explain the linear ordering on braids found by Dehornoy.
am beginning to add some genuine content to Ruelle zeta function (which used to just redirect to zeta function of a dynamical system which in turn is no more that a stub)
not done yet
Started a general page about beta reduction, and a stub for eta reduction but I have to run now.
I had expected that we had this entry already, but it seems we did not: Spanier-Whitehead category
Stub for semantics of a programming language.
Began cotopological localization.
I have added Zurek’s reference
which is surveying the decoherence approach to the collapse of the wave function and the einselection to interpretation of quantum mechanics.
The very first reference in the list is at a link at Perimeter institute which is outdated.
I created a stub for object of finite length to get rid of a grey link and then one for composition series.
A stub for Rees matrix semigroup.
I just added to abelian group and to compact closed category the basic but unmentioned fact that an abelian group can be seen as a discrete compact closed category.
recent activity made me feel it is time for a new context-cluster floating-table-of-contents: started:
and included it as a floating TOC into relevant entries.
(The table itself clearly should be expanded and better organized much more. But it’s maybe a start.)
I have added to (infinity,2)-sheaf a section Examples - Codomain fibration / canonical (∞,2)-sheaf with the statement that for 𝒳 an (∞,1)-topos, the ∞-functor
A↦𝒳/Ais always an (∞,2)-sheaf with respect to the canonical topology.
(It’s the (∞,2)-sheaf of “unstable quasicoherent (∞,1)-sheaves”!)
Added the statement of Urysohn’s lemma. The stub that had been there listed only a link, which is now rotten (it was to some lecture notes), so I added a planetmath link which ought to be stable.
basic definition at Thom collaps map
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’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 Ab, 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 Spec(𝕊).
Just because as an nLab 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 Spec(MU).
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 nLab 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 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 n-stack at any point is a formal moduli problem, hence equivalently an L∞-algebra. Made the corresponding remark more explicit also at cohesive (∞,1)-presheaf on E-∞ rings.
New stub stochastic quantization.
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 hTop with hTop*, and have included a reference for the claim that there are cogroups in hTop* 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.
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 nLab 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
| 0≤k≤n | transgression to dimension k |
|---|---|
| 0 | extended Lagrangian, universal characteristic map |
| k | (off-shell) prequantum (n-k)-bundle |
| n−1 | (off-shell) prequantum circle bundle |
| n | action functional = prequantum 0-bundle |