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stub for integral transform (also a highly stubby stub for Fourier-Mukai transform)
started stub entry sheaf of L-infinity algebras, but it is still lacking some evident references
Originally I was going to add a comment on how to axiomatize in differential cohesion a sheaf of -algebras over as a pointed object in which is sent by the reduction modality to an identity. But maybe I’ll better do this tomorrow, when I am more awake (or else whenever that happens again).
I have started an entry on rewriting. It is just a stub for the moment.
Added to natural number a discussion about the fact that constructively, the natural numbers may fail to be (order) complete, as highlighted by Andrej Bauer in a very nice blog post. I quite like this example, because by interpreting a related lemma in the internal language of a certain sheaf topos one obtains a well-known proposition in algebraic geometry almost for free (see entry); but please let me know if stuff like this is too localized for the nLab.
created focal point
I typed at local topos in the section Local over-toposes statement and poof that sufficient for a slice topos to be local is that is tiny .
What are necessary conditions? Is this already necessary?
I added a few words to several complex variables, even though I am out of my depth here. If we have analysts popping by here, hopefully they will get an urge to add more.
stub for Anaxagoras’s Nous.
I wrote analytic function, mostly just a definition. I found a reference that treated the infinite-dimensional case in pretty fair generality (slightly more than I actually did) without making the definition any more complicated (well, except one place where one must insert the word ‘continuous’), so I did that.
I mentioned the intermediate value theorem at pentagon decagon hexagon identity and then began an article on it.
created differentiation and chain rule
brief entry complemented lattice, just to satisfy the link at quantum logic
I added a reference to C-star-system. I propose that we change the name of the page to the C-star dynamical system; this is the standard full term, jargon which is skipping dynamical is confusing for an outsider and not explicative. I can imagine many other things which deserve that name.
I have cross-linked de Morgan duality with Wirthmüller context for the statement that in linear logic
Also I have tried to make more of the links in the tables at de Morgan duality point to something.
The convention, when describing ring extensions, everywhere I’ve seen a convention, is that
I have adjusted four instances of former “at” on three pages that would be, algebraicwise, “away from” (and so they now appear).
Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing at weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object -enriched category with morphisms “multiply-by”, the localization-of-the-category “at ” (or its -enriched version, if saying that is necessary) really means the localization-of-the-ring “away from ”.
You all can sort out that contravariance as/if you like, but don’t break the old algebra papers!
created a brief entry membrane matrix model with some commented pointers to the literature
Added to field examples of internal fields: the canonical ring objects of the petit resp. gros Zariski toposes of a scheme.
Started an entry on closed morphisms, containing examples and characterizations using the internal language. Then I noticed that an entry on closed map already exists, but at the moment the nLab is too slow for proper browsing and editing. Will finish later and maybe merge the entries.
I repaired the definition of “unramified morphism” of schemes.
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)
added some content to tangent cone
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 an atom in a cohesive -topos over , then also the slice 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!)