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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory itex k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure measure-theory modal-logic model model-category-theory monad monads monoid monoidal monoidal-category-theory morphism motives motivic-cohomology nlab nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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- Discussion Type
- discussion topicGrothendieck duality
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by hilbertthm90
- Last Active Sep 1st 2011

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.

- Discussion Type
- discussion topicHadwiger's theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 1st 2011

stub for Hadwiger’s theorem

- Discussion Type
- discussion topicsymplectic geometry - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 31st 2011

started a contents-page symplectic geometry - contents and added it as a floating toc to relevant entries (there should be more, have not included everything yet)

- Discussion Type
- discussion topichigher Kleinian- and Cartanian geometry
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 31st 2011

started entries

collecting some evident ideas and observations, most of which we have discussed at one point or other over at the $n$Café.

- Discussion Type
- discussion topicrigid body dynamics
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by zskoda
- Last Active Aug 31st 2011

prompted by the discussion in another thread I have started creating stubs

But not much real content there so far.

- Discussion Type
- discussion topicinstanton Floer homology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Aug 30th 2011

I am starting an entry instanton Floer homology branched off from Floer homology

- Discussion Type
- discussion topicquantum group Fourier transform
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Aug 30th 2011

New entry quantum group Fourier transform (and related stubs coregular action, dual bialgebra, S. L. Woronowicz).

- Discussion Type
- discussion topicsymbol map
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 30th 2011

created symbol map and added a sentence to this effect to the Properties-section at Clifford algebra

- Discussion Type
- discussion topicSupergravity and superstrings
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by zskoda
- Last Active Aug 30th 2011

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!

- Discussion Type
- discussion topiccocommutative coalgebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Aug 29th 2011

created cocommutative coalgebra

- Discussion Type
- discussion topicHamiltonian dynamics on Lie groups
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 29th 2011

- Discussion Type
- discussion topicstrict Lie 2-algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Aug 29th 2011

created stub for strict Lie 2-algebra

- Discussion Type
- discussion topictrees as terminal coalgebra
- Category Latest Changes
- Started by Todd_Trimble
- Comments 28
- Last comment by TobyBartels
- Last Active Aug 28th 2011

I added some discussion at terminal coalgebra that the category of trees (equivalently, the category of forests $Set^{\omega^{op}}$) 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 $\omega^{op} \to Set$. 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!).

- Discussion Type
- discussion topicvacuum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 27th 2011

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.

- Discussion Type
- discussion topicmeasure coalgebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 27th 2011

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

- Discussion Type
- discussion topicMorse homology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 26th 2011

just for completeness, some references at Morse homology

- Discussion Type
- discussion topicdilaton gravity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 26th 2011

added to dilaton the action functional of

*dilaton gravity*(Perelman’s functional)Also references and maybe something else, I forget.

- Discussion Type
- discussion topictemporal gauge
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 26th 2011

unmotivated stub for temporal gauge (which I happened to need at Yang-Mills instanton and instanton Floer homology)

- Discussion Type
- discussion topicCharacteristic forms and geometric invariants
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 26th 2011

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.

- Discussion Type
- discussion topicstring theory -- contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 24th 2011

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.

- Discussion Type
- discussion topicfivebrane
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Aug 24th 2011

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.

- Discussion Type
- discussion topicVolume pseudoforms
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Aug 24th 2011

I wrote about these on Hodge star and volume form.

- Discussion Type
- discussion topicopinions on development of mathematics
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by TobyBartels
- Last Active Aug 24th 2011

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.

- Discussion Type
- discussion topicconnection on a 2-bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 23rd 2011

At connection on a 2-bundle I have started an Examples-section and added References. Both long overdue.

- Discussion Type
- discussion topicreal closed field
- Category Latest Changes
- Started by Todd_Trimble
- Comments 19
- Last comment by Mike Shulman
- Last Active Aug 23rd 2011

I added a little bit of material to ordered field, namely that a field is orderable iff it is a real field (i.e., $-1$ is not a sum of squares). More importantly, at real closed field, I have addressed an old query of Colin Tan:

Colin: Is it true that real closure is an adjoint construction to the forgetful functor from real closed fields to orderable fields?

by writing out a proof (under Properties) that indeed the forgetful functor from category of real closed fields and field homomorphisms to the category of real fields and field homomorphisms has a left adjoint (the real closure). Therefore I am removing this query from that page over to here.

- Discussion Type
- discussion topicconstructible
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by Mike Shulman
- Last Active Aug 23rd 2011

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.

- Discussion Type
- discussion topic2-pullback
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by TobyBartels
- Last Active Aug 22nd 2011

Query from Stephen on 2-pullback.

- Discussion Type
- discussion topictorsion of a metric connection
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 22nd 2011

some quick content at torsion of a metric connection

- Discussion Type
- discussion topicisometry group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 22nd 2011

stub for isometry group

- Discussion Type
- discussion topicProper homotopy theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 13
- Last comment by Andrew Stacey
- Last Active Aug 22nd 2011

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!)