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stub for metaplectic structure
added a few more Examples to reduction of structure groups.
created metalinear structure. Added it to square roots of line bundles - table . Linked to it from Theta characteristic and so forth.
started metalinear group
Danny Stevenson was so kind and completed spelling out the proof of the pasting law for -pullbacks here at (infinity,1)-pullback.
I created a stub for Kirchhoff’s laws to go with the Café-discussion here. Maybe somebody feels like expanding it, I don’t really have the time for this right now.
I have created an entry symplectic reduction - table and included it into relevant entries.
I wrote Hamiltonian action.
I tried to say precisely what the action is by. In the literature (but also in a previous version of our moment map entry) there is often (for instance on Wikipedia, but also in many other sources) an imprecise (not to say: wrong) statement, where an action by Hamiltonian vector fields is not distinguished from one by Hamiltonians.
I have decided to splitt off a stand-alone entry symplectic reduction from BRST-BV formalism (which used to be the redirect). Still just a stub. Lots of material and references still needs to be copied or moved from the latter to the former.
I have started a table geometric quantization - contents and added it as a floating TOC to the relevant entries.
Parts of this remain a bit unfinished. The Lab is pretty much unusable in the last hours. I’ll give up now, have wasted too much time with this already. Maybe later it has recovered.
I have created a table geometric quantization extensions - table.
Mostly I have been editing aspects of the entries listed in this table here and there. Also included the table in the Properties-section of various of these entries.
I have started a table of contents representation theory - contents (based on the link list at representation theory) and started adding it as a floating table of contents to relevant entries.
But I ran out of steam before being entirely satisfied with the result.
stub for IHL manifold.
I realized that infinite-dimensional manifold and all entries related to it are still very stubby.
I am not attempting to change this now, but I thought a first step to make progress is to list what stubs we actually have. I came up with this list
and added it to manifolds and cobordisms - contents and made sure that all these entries point to each other.
Now somebody go and add more content to these entries! :-)
In case you see the activity in the logs and are wondering, I should say that I have been working on a new entry higher geometric quantization (that used to redirect to n-plectic infinity-groupoid).
I have started adding some survey-tables. But not done yet with the entry as a whole.
I felt it was time for another table: homotopy-homology-cohomology
The structure is just a first attempt, begun in a brief moment of leisure. I’ll try to think about how to improve on it. Let me know what you think.
I have started to include this into relevant entries.
started an entry associated infinity-bundle
in order to summarize the thesis by Matthias Wendt on associated -bundles in arbitrary -toposes, generalizing the classical old results by Stasheff and May from .
Also added some remarks on the relation to the discussion at principal infinity-bundle. Hopefully to be continued tomorrow.
I am now going through the section Structures in a cohesive oo-topos and polish and expand the discussions there.
First thing I went through is the subsection Geometric homotopy and Galois theory. It gives the definition of the fundamental -groupoid functor, a proposition on its consistency (which we had mentioned elsewhere), the definition of locally constant -stacks in the sense of -principal -bundles, and then the central theorem of Galois theory, proven by applying the -Yoneda lemma iteratively.
(This is material appearing in one form or other in other entries and at this point does not invoke the -locality, but I want to have here all in one place a nice comprehensive discussion of the whole situation in a cohesive -topos.)
Stubs (just to provide the links to the basic references so far) for Nekrasov functions, AGT conjecture, and, less related, spectral networks.
I created minimal fibration which could be merged with minimal Kan fibration. The idea-section says that this notion is needed to give a well defined notion of n-category. However there are other applications which I didn’t mention.
started a stub for ring extension to go with the Café-discussion here.
created carrying
just heard an interesting talk by Steven Rosenberg on CS invariants on infinite-dimensional manifolds. So I created an entry infinite-dimensional Chern-Simons theory in order to record some references
and handful of further details at over quasi-category
I made redirects to Online Resources, namely the math blogs, online resources. Before we were complaining to Online Resources for many reasons including that it is not of all resources but only of blogs and wikis in relevant areas. No list of main institutes and archives like arXiv, numdam, jstor etc. there. As the list is long, and hard to scroll, I suggest not to add those to the current page. I think we should rename the current page to math blogs eventually and keep Online Resources (especially because of John's reference in his AMS Notices paper) as a redirect and create new pages for other stuff as well as organize the whole system around a top page math resources which will link to math blogs, math archives, math institutions (and maybe more) as well as very comprehensive central AMS-kept list of math resources.
I know it is not only about math here, but math is a short abbreviation for page name.
Up to now I have realized a large part of an above program, see math archives, math institutions and the supposed top resource page math resources, except that I was cautious not to rename the page Online Resources as people may disagree even with keeping the old redirect and because it may be tricky with the cache bug, while the page is of central importance. I think it would be useful if the pages like math institutions and the top page math resources stay not much longer than they are now, to have quick links and nice readability/visibility. This is the most effective organization, I think. For smaller institutions societies and alternative small lists of resources, it is better to go via links at AMS, EMS and IMU which are already efefctively linked. We can not do better there than those societies do, apart from listing few extra main resources of our main interest. We can have a separate page just for categories or some other things. But the list of blogs is of different character, unlike going to AMS page or jstor, one does not need to go that quickly through list of less-organized stuff like blogs. So the blog list math blogs should grow indefinitely...I have chosen plural as before in these pages, without singular redirect at the moment.
Added the reference
Maria Emilia Maietti, Modular correspondence between dependent type theories and categories including pretopoi and topoi, Math. Struct. in Comp. Science (2005), vol. 15, pp. 1089–1149
I am starting fiber integration in ordinary differential cohomology, but just two references so far (are there more? the project by Baer seems not to have borne any article, it seems)
I have created energy ex nihilo. Take that, Hermann von Helmholtz!
I started display of a p-divisible group.
I may have written something at Kervaire invariant, but it is at best a stub for the moment
I happened to have need for a stub Born-Oppenheimer approximation
stub for Kac-Moody group, in order to record some references
stub for formal loop space, just so as to record Kapranov’s article. Eventually it would be great if the Lab had some paragraphs on the geometric link with the chiral de Rham complex, vertex operator algebras, etc. But I can’t do it now.
Some reorganization and added material at type theory. In particular, I added some of the basic syntax of type theories, and also some comments about extensional vs. intensional type theories.
Stubs for elementary class of structures (with redirect elementary class) and a stub for far more general Saharon Shelah’s notion of an abstract elementary class.
Stub for amalgamation.
We already have the entry predicate calculus (or first-order logic); I have created separate stub first-order theory, almost empty now, and which could just have been redirect to first-order logic, though I think eventually it would be good to have them separate, as under first-order theories version one could list lots of standard examples of first-order theories, what does not really fit into predicate logic entry, where one should really deal more with predicate calculus. PLus one should do other views of first-order theories.
created Dixmier-Douady class
Stub constructible set, not yet precise (e.g. the universe is not a Boolean algebra as it is a proper class), but gives idea what we could work on in the entry.
Changes to dynamical system. Stub for algebraic dynamics.
New entry structure but the Lab is down so I save here the final version of editing, which is probably lost in Lab:
The concept of a structure is formulated as the basic object of mathematics in the work of Bourbaki.
In model theory, a structure of a language is the same as model of with empty set of extra axioms. Given a first-order language , which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including ) and quantifiers; a structure for , or -structure is a set with an interpretation for symbols:
if is an -ary relation symbol, then its interpretation
if is an -ary function symbol, then is a function
if is a constant symbol, then
Interpretation for an -structure inductively defines an interpretation for well-formed formulas in . We say that a sentence is true in if is true. Given a theory , which is a language together with a given set of sentences in , the interpretation in a structure makes those sentences true or false; if all the sentences in are true in we say that is a model of .
Some special cases include algebraic structures, which is usually defined as a structure for a first order language with equality and -relation both with the standard interpretation, no other relation symbols and whose function symbols are interpreted as operations of various arity. This is a bit more general than an algebraic theory as in the latter, one needs to have free algebras so for example fields do not form an algebraic theory but are the algebraic structures for the theory of fields.
In category theory we may talk about functor forgetting structure (formalizing an intuitive, related and in a way more general sense), see
I added in the definition of algebraic group the requirement ”field” into ”algebraically closed field”. Alternatively one could omit ”field” in the definition at all since this is implicit in ”variety”.
I took some notes during my reading of (Chapters I-III of)
In the recent days I inserted parts of these notes in different nlab entries. Maybe it is of use to somebody to have all of these notes in wiki-form. So I created lectures on p-divisible groups containing the skeleton of the contents. I will fill in the parts I have written so far (roughly chapters I and II) tomorrow. Of course anyone should feel free to rewrite or complete the related entries. Currently the page names contain the chapter-numbering from the original text - I think this numbering can be discarded at the time the linked page contains more information than the original text. If there is some nlab policy on wiki-ed texts suggesting otherwise, please let me know.
Finally, definable set.
I created locally bounded category with basic results from papers of Kelly and Lack. My motivation (unfortunately not reflected in the current stub) is to provide a reference for convergence conditions for the free monad construction.
On this topic, does anyone know whether there are reasonable conditions under which the dual “free comonad” construction would converge? I’m concerned by the result at locally presentable category (new to me; does anyone have a reference?) which says that the opposite of a locally presentable category is locally presentable only if the category is a poset.
I have expanded theory adding more basics in classical syntactic approach. I added a new subsection
Set-theoretic models for a first-order theory in syntactic approach
The basic concept is of a structure for a first-order language : a set together with an interpretation of in . A theory is specified by a language and a set of sentences in . An -structure is a model of if for every sentence in , its interpretation in , is true (“ holds in ”). We say that is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for (in our universe). Two theories, , are said to be equivalent if they have the same models.
Given a class of structures for , there is a theory consisting of all sentences in which hold in every structure from . Two structures and are elementary equivalent (sometimes written by equality , sometimes said “elementarily equivalent”) if , i.e. if they satisfy the same sentences in . Any set of sentences which is equivalent to is called a set of axioms of . A theory is said to be finitely axiomatizable if there exist a finite set of axioms for .
A theory is said to be complete if it is equivalent to for some structure .
I have started an entry analytic space with material on Berkovich’s non-rigid analytic geometry.
I don’t really know this subject and have been adding material to the entry as I read about it and to the extent that I correctly understood it. Experts are most welcome to help out.
As indicated here, I am motivated by the following: Berkovich’s local contractibility result suggests that the -topos of -sheaves over the site of -adic analytic spaces might be cohesive.
The idea would be that his result implies (if it does) that the site (category with coverage) of contractible -adic afine spaces is a dense subsite of that of all -adic spaces. Since it should be an infinity-cohesive site that would imply the claim.
But despite looking through Berkovich’s writings for a little bit today, I am still not sure if he just shows that the underlying topological space of a -adic anayltic space is locally contractible, or if one may indeed deduce that they are locally contractible with respect to étale homotopy, as would be needed for the above conclusion.
Stub for Lascar group, the analogue of Galois group for first order theories.
created generalized vielbein
I added a definition-section to formal scheme with the four equivalent definitions of a k-formal scheme from Demazure, lectures on p-divisible groups. There is some overlap with the section on Noetherian formal schemes now.
created stubs for
This is not supposed to be satisfactory content. I just wanted these pages to exist right now, so that links to them work.
I treid to clean up Whitehead tower a bit:
I rewrote and expanded the Idea/Definition part.
Then I moved David Roberts' material that was there to the appropriate section at the new Whitehead tower in an (infinity,1)-topos. (There I tried to add some introductory remarks to it but will try to further highlight David's results here in a moment).
At Whitehead tower I left just a new section that says that there is a notion of Whitehead towers in more general contexts with a pointer to Whitehead tower in an (infinity,1)-topos
felt like creating double cover
felt like creating generalized tangent bundle
Uday has added :
Mac Lane, VII.4, only requires a monoidal category to define actions. – at action.
This takes up an old point that I made but never felt up to following up.