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I noticed that the two links : André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf), near the bottom of the entry join of quasi-categories are dead. Does anyone have a more recent link? or if not an alternative reference?
have added some brief Idea-section and lists of references to
Starting a stub Fredholm determinant.
I have added the following reference to Berkovich space. Judging from the abstract this sounds like I nice unifying perspective. But I haven’t studied it yet
We show that Berkovich analytic geometry can be viewed as algebraic geometry in the sense of Toën-Vaquié-Vezzosi over various categories. The objects in these categories are vector spaces over complete valued fields which are equipped with additional structure. The categories themselves will be quasi-abelian and this is needed to define certain topologies on the categories of affine schemes. We give new definitions of categories of Berkovich analytic spaces and in this way we also define (higher) analytic stacks. We characterize in a categorical way the G-topology or the topology of admissible subsets used in analytic geometry. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the categories which we introduce. We also include a treatment of quasi-coherent sheaf theory in analytic geometry proving Tate’s acyclicity theorem for quasi-coherent sheaves. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.
I wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry
This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story.
(Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the Lurie spectral sequence.)
I want to further fine-tune this. But not tonight.
created an entry simplicial object in an (infinity,1)-category and interlinked it a bit. Nothing much there yet, for the moment this is mostly a reminder for me to get back to it later.
am starting spectral sequence of a simplicial stable homotopy type, but right now it’s just a stub.
Have expanded the Lurie spectral sequences – table further:
added a tad more content to infinity-Dold-Kan correspondence
stub for Einstein-Hilbert action
as you may have seen in the logs, I am working on an entry Higher toposes of laws of motion, something like extended talk notes.
I am running a bit out of time, and so the entry is unpolished and turns into just a list of keywords towards the end, for the moment. But in case anyone is wondering about the logs, here is the announcement.
Don’t look at this yet if you feel like just reading. Of course if you feel like joining in with the editing a bit, that’s welcome, as usual.
The characterization of formally étale morphisms of schemes by the infinitesimal shape modality had been scattered a bit through the nLab (at Q-category, at formally etale morphism a bit, at differential cohesion a little).
To make the statement more recognizable, I put it into this new entry here:
I split off an entry dg-geometry from the entry on Hochschild geometry, since it really deserves a stand-alone discussion.
Eventually somebody should add the references by Kapranov et al on dg-schemes etc. And much more.
started topos of laws of motion (lower case!) on the actual notion as such.
created a brief entry rational thermodynamics.
I haven’t actually seen yet the actual detail of this axiomatics (but see the citations given at the above link). What I currently care about is this historical fact, which I added to the Idea section:
What is called rational thermodynamics is a proposal (Truesdell 72) to base the physics of irreversible thermodynamics on a system of axioms and derive the theory from these formally.
The success of the axioms of rational thermodynamics as a theory of physical phenomena has been subject of debate. But the idea as such that continuum physics can be and should be given a clear axiomatic foundation seems to have inspired William Lawvere (see there for more), once an undergraduate student of Clifford Truesdell, to base continuum mechanics on constructions in topos theory, such as synthetic differential geometry and cohesion.
stub for infinitesimal extension (and cross-linked a bit)
Zoran: sorry, I know I should cite that article of yours/your colleagues, could you please add it? Thanks.
as last week, I have created an entry that collects some of the recent edits scattered over the nLab supposedly in one coherent story, it’s
Should be expanded a bit more. But not tonight.
At the entry classifying morphism, there is a query (not displayed as one):
Where can I find a construction of the classifying morphism to a classifying space for a G-bundle with connection - using the connections as a 1-form?
Given the subject matter and the name ‘jim’ plus the location of 96.245.205.76 (Pennsylvania …)????
With Zoran I am working on entries related to monadic descent. While that is underway, I have added to
a section
and
at Artin-Schreier sequence I spelled out the existence proof. This derserves/demands to be further polished and streamlined, but i’ll leave it at that for the moment.
added to open immersion of schemes brief mentioning of the relation to etale morphisms and to Zariski opens
I have created a stub for the acyclic assembly lemma as there was a grey link in another entry.
in my search for a good way of introducing basics of étale cohomology I switched from Milne to Tamme, and started some hyperlinked index for the latter’s Introduction to Étale Cohomology. As before, in the course of this I created some brief entries for keywords there, if they didn’t exist yet.
Hope to expand this now…
Fixed a couple incorrect statements at hypercomplete (infinity,1)-topos:
some trivial/stubby edits, announced here just in case anyone is wondering about edit activity:
added more references to étale (∞,1)-site
added a tad more text to the stubs Weil cohomology, étale cohomology
created stub l-adic cohomology
started splitting off formally étale morphism of schemes from formally étale morphism
I have started a hyperlinked index for Milne’s Lectures on Étale Cohomology
(Up to and including section 27, where the proof of the Weil conjectures starts. )
In the course I have created a bunch of brief entries, if the corresponding keyword didn’t yet have one. Also cross-linked vigorously.
added to the list of equivalent conditions in the definition at étale morphism of schemes the pair “smooth+unramified”. Added a remark after the definition on how to read these pairs of conditions.
brief note comparison theorem (étale cohomology)
brief entry complex analytic topology, just for completeness
we had an entry spectrum (geometry) which wasn’t linked to from almost anywhere, in particular not from spectrum - disambiguation.
I have now added a bunch of cross-links, between these two entries and between the entries that they link to. It’s better now than it was before, but could still do with further improvement.
the entry profinite space and entries related to it were/are a bit in need of some care.
The entry used to start out saying “Profinite space is another word for Stone space”, which was misleading, because there is a not-entirely-trivial equivalence involved. So I changed it to
A profinite set is a pro-object in FinSet. By Stone duality these are equivalent to Stone spaces and thus are often called profinite spaces. So these are compact Hausdorff totally disconnected topological spaces.
Also cross-links with profinite reflection were missing, and so I added them. Also cross-linked with finite set, with localic reflection and maybe with more.
The entry profinite space is still stubby/unsatisfactory.
started entries
Question: We have the implications
étale morphism ⇒ weakly étale morphism ⇒ formally étale morphism
but can one say more specifically what kind of generalized finite presentability condition makes a formally étale morphism a weakly étale morphism?
I have fixed some dead links relating to Loday which were occurring n several pages.
made a note of a simple observation:
for X an atom in a cohesive ∞-topos H over ∞Grpd, then also the slice H/X is cohesive … except possibly for the property that shape preserves binary products (but it does preserve the terminal object):
stub for chromatic spectral sequence, so far mainly to record that it arises as the Lurie spectral sequence of the chromatic tower (thanks to Dylan Wilson and Marc Hoyois)
Created continuous algebra.
now the preprint referred to at tangent cohesive (infinity,1)-topos is out:
I noted an entry on generalized Eilenberg-MacLane spaces, but note that there is another use of this term in the literature, namely the representing fibrations for cohomology with local coefficients. These are the fibrations used by Gitler and then by Alan Robinson, Hans Baues and others more recently. What would be the preferred name for these latter things. (I personally find the idea of giving a name to products of Eilenberg- Mac Lane spaces other that ‘products of Eilenberg - Mac Lane spaces’ a bit strange, but I know that there is some strange terminology around!)
I noticed by accident that we have an entry coinvariant. Then I noticed that we also have an entry homotopy coinvariant functor.
I have now added cross-links between these entries and with invariant and orbit, so that they no longer remain hidden.
I also edited the first case of group representation coinvariants at coinvariant a little.
this Physics.SE question made me create a category:reference entry for
I noticed that some old entries were requesting a keyword link for brave new algebra, so I created it and filled in a default-paragraph. Please feel invited to expand.
In that context I have a question: the dual generlized Steenrod algebras have been called “brave new Hopf algebroids” in articles including
Andrew Baker, Brave new Hopf algebroids (pdf)
Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of MU-algebras, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (Euclid)
Mark Hovey, Homotopy theory of comodules over a Hopf algebroid (arXiv:math/0301229)
But the Hopf algebroids considered in these articles are ordinary Hopf algebroids, they are given not by Hopf ∞-algebras (E,E∧E) but by their homotopy groups (E•,E•(E)), unless I am missing something.
So at least without further discussion, calling (E•,E•(E)) “brave new” is a bit of a stretch. The brave new thing would be (E,E∧E) (if indeed it is a “Hopf ∞-algebroid”).
Can anyone say more about this? I can’t seem to find any source talking about this. The canonical guess of googling for “derived Hopf algebroid” doesn’t show relevant results.
created Landweber-Novikov theorem
started motives in physics with text that I posted as an answer to this Physics.SE question.
Needs to be polished and expanded. But I have to run now.
since it appears in several entries and probably more to come, I gave it an entry of its own: Quillen’s theorem on MU
also gave an entry to Lazard’s theorem
I needed finite spectrum
To be able to conveniently link I have also splitt of finite CW-complex from CW-complex
some basics at Lubin-Tate theory
(wanted to do more, but the nLab is giving me a really hard (down-)time )
created a table-for-inclusion image of J – table listing pertinent information in low degree, and included it in some relevant entries
started a very stubby
along with very stubby
and a very stubby
Had wanted to do more, but now I am running out of steam. Maybe the stubs inspire somebody to add a little more…
created a brief entry for Bousfield equivalence
I messed up slightly: i had forgotten that there was already a stub titled contact geometry. Now I have created contact manifold with some content that might better be at contact geometry. I should fix this. But not right now.
created Einstein’s equation, only to record a writeup by Gonzalo Reyes which I just came across by chance, who gives a discussion in terms of synthetic differential geometry.
quick entry for phantom map
brief Idea-section at chromatic convergence theorem
stub for telescopic localization,
finally created the category:reference-entry for Lurie’s chromatic lecture. See Chromatic Homotopy Theory
(And as a special service to the community… with lecture titles. ;-)
Lecture 1 Introduction (pdf)
Lecture 2 Lazard’s theorem (pdf)
Lecture 3 Lazard’s theorem (continued) (pdf)
Lecture 4 Complex-oriented cohomology theories (pdf)
Lecture 5 Complex bordism (pdf)
Lecture 6 MU and complex orientations (pdf)
Lecture 7 The homology of MU (pdf)
Lecture 8 The Adams spectral sequence (pdf)
Lecture 9 The Adams spectral sequence for MU (pdf)
Lecture 10 The proof of Quillen’s theorem (pdf)
Lecture 11 Formal groups (pdf)
Lecture 12 Heights and formal groups (pdf)
Lecture 13 The stratification of ℳFG (pdf)
Lecture 14 Classification of formal groups (pdf)
Lecture 15 Flat modules over ℳFG (pdf)
Lecture 16 The Landweber exact functor theorem (pdf)
Lecture 17 Phanton maps (pdf)
Lecture 18 Even periodic cohomology theories (pdf)
Lecture 19 Morava stabilizer groups (pdf)
Lecture 20 Bousfield localization (pdf)
Lecture 21 Lubin-Tate theory (pdf)
Lecture 22 Morava E-theory and Morava K-theory (pdf)
Lecture 23 The Bousfield Classes of E(n) and K(n) (pdf)
Lecture 24 Uniqueness of Morava K-theory (pdf)
Lecture 25 The Nilpotence lemma (pdf)
Lecture 26 Thick subcategories (pdf)
Lecture 27 The periodicity theorem (pdf)
Lecture 28 Telescopic localization (pdf)
Lecture 29 Telescopic vs En-localization (pdf)
Lecture 30 Localizations and the Adams-Novikov spectral sequence (pdf)
Lecture 31 The smash product theorem (pdf)
Lecture 32 The chromatic convergence theorem (pdf)
Lecture 33 Complex bordism and E(n)-localization (pdf)
Lecture 34 Monochromatic layers (pdf)
Lecture 35 The image of J (pdf)
created a stub for cluster decomposition, since I wanted the link elsewhere, but nothing there yet…
I recently created entry Bol loop. Now I made some corrections and treated the notion of a core of a right Bol loop (the term coming allegedly from Russian term сердцевина).
created a disambiguation page: spectral geometry
New article: Tychonoff space.