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I’ve moved part of my beginner’s summery on Mac Lane’s proof of the coherence theorem for monoidal categories to the nLab.
Surely there are many mistakes, possibly fatal: I am most worried about the naive definition I used for the syntax of arrows (sadly, I could only use my nearly zero knowledge of logic), and the part including the units and unitors. There are probably many itex errors too.
At some point I’ve realized that it was silly to use the cumbersome arrow language as is (say, writing a string like αuvu instead of αu,v,w, etc.), but to correct this required changing all figures, and this is too much for me at the moment. I also apologize for using files for figures (admiting that some of them shouldn’t even be figures), but it was too much work to do all the transition from my notes to itex in one jump.
I hope learning by seeing this page modified by people knowing more than me (anyone here, that is :) ), but if this page seems beyond hope, I have no objections to renaming it so that it will not clutter the ’lab.
By the way, now that I’ve re-entered the page, all figures appear with a question mark, and to the see the figure I have to press on the question-mark link (previously I’ve seen all figures appearing in the page). Is there any way to solve this problem?
To add recent surge of activity (by Urs, me etc.) in nLab on symplectic geometry, variational calculus and mechanics I created the entry Lagrange multiplier following mainly Loomis-Strenberg. For convenience, I uploaded the critical 4 pages from their book.
I am starting an entry symplectic infinity-groupoid.
This is still in the making. Currently there are two things:
A little general indication of what this is supposed to be about;
A proof of an assertion that serves to justify the whole concept.
Namely, the literature already knows the concept of a symplectic groupoid. This plays a big role in Weistein’s program and in particular in geometric quantization of symplectic groupoids, which induces, among other things, a notion of geometric quantization of Poisson manifolds.
As far as I am aware (though I might not have been following the latest developments here, would be grateful for comments) it is generally expected that symplectic groupoids are formally the Lie integration of Poisson Lie algebroids, but there is no proof or even formalization of this in the literature.
In the entry I indicate such a formalization and give the respective proof.
The idea is that this is a special case of the general machine of infinity-Chern-Weil theory:
namely: the symplectic form on a symplectic Lie n-algebroid such as the Poisson Lie algebroid is Lie theoretically an invariant polynomial. So the ∞-Chern-Weil homomorphism produces a corresponding morphism from the integrating smooth ∞-groupoid to de Rham coefficients. This is a differential form in the world of smooth ∞-groupoids.
The assertion is: this comes out right. Feed a Poisson Lie algebroid with its canonical invariant polynomial into ∞-Chern-Weil theory, out comes the “classical” symplectic Lie groupoid.
(I do this for the case that the Poisson manifold is in fact itself symplectic, which is the only case I remember having seen discussed in earlier literature. But I think I can generalize this easily.)
started something at higher symplectic geometry.
Not much there yet, more a kind of announcement for the moment. But it may serve the purpose of providing a sentence on in which sense multisymplectic geometry and symplectic Lie n-algebroids are two aspects of a more general thing.
I addede a paragraph at Poincaré duality about the generalizations, and created the entry (so far only descent bibliography) Grothendieck duality; the list of examples expanded at duality. All prompted by seeing the today’s arXiv article of Drinfel’d and Boyarchenko.
stub for Hadwiger’s theorem
started entries
collecting some evident ideas and observations, most of which we have discussed at one point or other over at the nCafé.
prompted by the discussion in another thread I have started creating stubs
But not much real content there so far.
I am starting an entry instanton Floer homology branched off from Floer homology
New entry quantum group Fourier transform (and related stubs coregular action, dual bialgebra, S. L. Woronowicz).
Can someone have a look at Supergravity and Superstrings - A Geometric Per? I don’t suspect this is really Spam but it is a strange entry!
created stub for strict Lie 2-algebra
I added some discussion at terminal coalgebra that the category of trees (equivalently, the category of forests Setωop) is a terminal coalgebra for the small-coproduct cocompletion (as endofunctor on Cat); this is a special case of Adamek’s theorem. I linked to this discussion by adding a section at tree. (There is of course closely related discussion at pure set as well.)
It seems to me that the nLab is a bit thin on general matters of recursion. I’ve been looking a bit at the discussion in Paul Taylor’s book, and I am becoming partial to the general idea that in some sense coalgebras and corecursion often come first; after that one may base recursive schemata on the notion of well-founded coalgebras. For example, (ill-founded) trees are really simple conceptually, or at least have a very simple uniform description: as presheaves ωop→Set. This is just a simple-minded application of Adamek’s theorem. Later, one can peek inside and gets the initial algebra for the small-coproduct completion as the category of well-founded trees, but this is by no means as simple (one can’t just apply Adamek’s theorem for constructing initial algebras – the hypotheses don’t hold here!).
wrote something at vacuum.
I mainly wanted the link to point somewhere. I don’t claim that what I have there presently is a good discussion. So I have labeled it “under construction” for the moment.
since it was mentioned on the category theory mailing list I went to the entry measure coalgebra and edited a bit: I have added some hyperlinks and Definition- and Proposition-environments.
Somebody who created the entry should look into this issue: currently the entry mentions a ground field right at the beginning, which however never reappears again. It’s clear that everything can be done over an arbitrary ground field, I guess, but currently this is not discussed well.
In order to satisfy links I then created
added to dilaton the action functional of dilaton gravity (Perelman’s functional)
Also references and maybe something else, I forget.
unmotivated stub for temporal gauge (which I happened to need at Yang-Mills instanton and instanton Floer homology)
The reference Characteristic forms and geometric invariants was missing from a few entries (for instance Chern-Simons theory and secondary characteristic class.) So I added it in. Since it is an important reference, I thought I should give it a “category: reference”-page of its own.
I shouldn’t be doing this. But in a clear case of procrastination of more urgent tasks, I created a floatic TOC string theory - contents and added it to some relevant entries.
I notice that in recent preprints (see equation (2.1) in today’s 1108.4060) people are getting awefully close to rediscovering nonabelian 2-connections in the worldvolume theory of NS-fivebranes (but they are forgetting the associator! :-).
This follows a famous old conjecture by Witten, which says that the worldvolume theory of a bunch of fivebranes on top of each other (what physicsist call a “stack” of fivebranes) should be a nonabelian principal 2-bundle/gerbe-gauge theory. If you have followed Witten’s developments since then (with his latest on Khovanov homology) you’ll know that he is suggesting that this theory is at the very heart of a huge cluster of concepts (geometric Langlands duality and S-duality being part of it).
So I should eventually expand the entry fivebrane . I’ll start with some rudiments now, but will have to interrupt soon. Hopefully more later.
I wrote about these on Hodge star and volume form.
I could not find a better title, for the new entry, unfortunately: opinions on development of mathematics (should be mainly bibiliography entry). I need some place to start collecting the titles which talk about generalities of mathematical development, what is important, what is not. This is relevant for but it is not philosophy. Not only because of traditional focus of philosophy on “bigger” things like true nature of beings, meaning, ethics, cognition and so on, but more because the latter is very opinionated in the usual sense, even politics. Though we should of course, choose those which have important content, it is useful to collect those. We can have netries like math and society, even math funding for other external things of relevance, eventually. This was quick fix as I have no time now.
At connection on a 2-bundle I have started an Examples-section and added References. Both long overdue.
I have created a stub for constructible universe. I did not go through the version of the definition via definability. Now constructible sets are sets in the constructible universe. The notion of course, intentionally reminds the constructible sets in topology and algebaric geometry as exposed e.g. in the books on stratified spaces, on perverse sheaves (MacPherson e.g.) and in Lurie’s Higher Topos Theory. Now I wanted to create constructible set but I was hoping that there is a common definition for all these cases or at least logically defendable unique point of view, rather than partial similarity of definitions. I mean one always have some business of unions, complements etc. starting with some primitive family, say with open sets, or algebraic sets, or open sets relative strata etc. and inductively constructs more. Now, all the operations mentioned seem to have sense in some class of lattices. Maybe in Heyting lattices or at least in Boolean lattices. On the other hand, google spits out several references on constructible lattices *one of the authors is certain Janowitz), but the definition there is disappointing. I mean I would like that one has some sort of constructible completion of certain kind of a lattice and talk about the constructible elements as the elements of constructible completion. I am sure that the nLab community could nail the wanted common generalization down or to give a reference if the literature has it already.
some quick content at torsion of a metric connection
stub for isometry group
I have started an entry on proper homotopy theory. This is partially since it will be needed in discussing some parts of strong shape theory, but it may also be useful for discussing duality and various other topics, including studying non-compact spaces in physical contexts. This is especially true for non-compact manifolds. (I do not know what fibre bundles etc. look like in the proper homotopy setting!)
A term that I see sometimes in discussions of foundations: ordinary mathematics.
New entry frame bundle. Correction at affine connection: affine connection is the principal GLn(k)-connection on the frame bundle of the manifold, not the connection on the tangent bundle, though the latter is a special case of the corresponding associated bundle connection. Urs, do you agree with the correction ? (I think that wikipedia, linked there, terminologically agrees).
Added another proof (this one not using the universal coefficients theorem) of the isomorphism Hn(M,U(1))≅U(1) for M a closed oriented n-manifold to Dijkgraaf-Witten theory.
At simplicial localization I have added in some basic definitions and some basic properties.
added remarks about the relation of Postnikov towers to coskeletons to
I created contents of contents. I find I’m losing sight of what’s in the nLab since there’s so much of it. Urs (and others) very helpfully go around putting in these “contents” links, but even then you have to be in a section to know that it’s there, so I slurped through the database and extracted all the “contents” pages and stuck them in a single page (via includes). It’s not sorted, but my idea is to update this from the database rather than revising it by hand.
It’s just a first idea at getting some sort of overview; I imagine that this sort of thing can be done much better with some sort of graph showing how the pages link together, but this was quick and easy.
New stubs absolute de Rham cohomology, L-function, prompted by one answer to my MathOverflow question and having just basic links. By the way, the link to the pdf file of a Kapranov’s article listed at de Rham complex does not seem to work.
I made a very, very brief start to K3 surfaces
New pages:
locally additive space: Something I’ve been musing on for a bit: inside all these “categories of smooth object” then we have the category of manifolds sitting as a nice subcategory, but that doesn’t give a very nice intrinsic definition of a “manifold”. By that I mean that suppose you knew a category of smooth spaces and took that as your starting point, could you figure out what manifolds were without knowing the answer in advance? “locally additive spaces” are an attempt to characterise manifolds intrinsically.
kinematic tangent space: Once out beyond the realm of finite dimensional manifolds, the various notions of tangent space start to diverge and so each acquires a name. kinematic refers to taking equivalence classes of curves. There’s a bit of an overlap here with some of the stuff on Frölicher spaces, but this applies to any (cartesian closed, cocomplete) category of smooth spaces.
Apart from a few little tweaks to do with wikilinks and entities, these were generated by my LaTeX-to-iTeX package. References and all.
am starting an entry simplicial Lie algebra
New entry descent of affine schemes: the fibered category of affine morphisms (SGA I.8.2 th.2.1) satisfies effective descent along any fpqc morphism. This fact is harder than the descent for quasicoherent sheaves of 𝒪X-modules.
The discussion about the finitary vs infinitary case at connected object made me realize that something analogous could be said about finitary vs infinitary extensive categories themselves. I added a remark along those lines to extensive category.
In response to a very old query at connected object, I gave a proof that in an infinitary extensive category C, that an object X is connected iff hom(X,−):C→Set merely preserves binary coproducts.
The proof was written in classical logic. If Toby would like to rework the proof so that it is constructively valid, I would be delighted.
I created Dieudonne module. What is the policy on accents. Technically this should be written Dieudonné module everywhere. There is a redirect. I also defined this in the affine case, but I’m pretty sure if you replace “affine” by “flat” everything should work still.
started unitary representation of the super Poincaré group – the super-analog of unitary representation of the Poincaré group – so far mainly in order to record some references.
Also created a stub super Poincaré group
I wrote Specker sequence, a topic in computability theory that also has applications to constructivism.
I’ve started the page Cartier module.
created an entry Bohrification
started an entry twisted spin structure. So far the main point is to spell out the general abstract definition and notice that this is what Murray-Singer’s “spin gerbes” are models of.
I added remarks on Cauchy completion to the Properties-section both at proset and poset.
Also made more explicit at poset the relation to prosets.
I notice that at proset there is a huge discussion section. It would be nice if those involved could absorb into the main text whatever stable insight there is, and move the remaining discussion to the nForum here.
I created the page Witt Cohomology.
Todd has added to Grothendieck topos the statement and proof that any such is total and cototal (and I have added to adjoint functor theorem the statement that this implies that all (co)limit preserving functors between sheaf toposes have (right)left adjoints).
I notice that we should really merge Grothendieck topos with category of sheaves. But I don’t have the energy to do this now.
I edited adjoint functor theorem a bit: gave it an Idea-section and a References-section and, believe it or not, a toc.
Then I opened an Examples-section and filled in what I think is an instructive simple example: the right adjoint for a colimit preserving functor on a category of presheaves.
added statement and pointer to the proof of the gravitational stability of Minkowski spacetime
I made a stubby start at unitary irreps of the Poincare group, titled this way to save space. Very eager to get to the bottom of things; this subject can't be that hard.
Happened to notice a question at bicartesian closed category.
Question: don’t you need distributive bicartesian closed categories to interpret intuitionistic propositional logic? Consider the or-elimination rule
Γ,A⊢CΓ,B⊢CΓ,A+B⊢CThe intepretations of the two premises will be maps of type Γ×A→C and Γ×B→C. Then the universal property of coproducts gets us to (Γ×A)+(Γ×B)→C, but we can’t get any farther – we need a distributivity law to get Γ×(A+B)→C.
stub for Killing vector
for the moment, out of laziness, I also made Killing spinor and covariantly constant spinor redirect to this