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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex 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 foundation 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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- Discussion Type
- discussion topicchain complexes over fields
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Nov 3rd 2011

I have added some remarks to chain complex, model structure on chain complexes, homotopy limit, and derivator regarding the fact that every chain complex over a field is equivalent to its homology (regarded as a chain complex with zero differentials).

- Discussion Type
- discussion topicbase (infinity,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 3rd 2011

In reaction to the public demand exhibited by Guillaume Brunerie's comments I have created an entry

- Discussion Type
- discussion topicDeformation retractions
- Category Latest Changes
- Started by TobyBartels
- Comments 8
- Last comment by Tim_Porter
- Last Active Nov 1st 2011

To replace some anonymous scribblings, I cribbed some definitions from Wikipedia to get a stub at deformation retraction.

- Discussion Type
- discussion topic2-framing
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 1st 2011

quick note on 2-framing

- Discussion Type
- discussion topicWISC
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by DavidRoberts
- Last Active Nov 1st 2011

I thought up until just a few minutes ago that I had proved that WISC was equivalent to local essential smallness of $Cat_{ana}$. Mike urged me to put my proof on the lab, but in doing so I discovered it was flawed. So now WISC just has a proof that the principle implies local essential smallness.

- Discussion Type
- discussion topicelectrodynamics
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 28th 2011

I noticed there is no entry electrodynamics so I “created” it as a redirect to the existing stub electromagnetism. Though I personally consider electromagnetism as a phenomenon in real world, while electrodynamics just as a theoretical formalism to describe it, i.e. a theory of electromagnetism. There is some overlap between existing entries, like there is another, rather stubby classical electrodynamics, and entry quantum electrodynamics. The real content is in electromagnetic field.

- Discussion Type
- discussion topic1-dimensional Chern-Simons theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 28th 2011

- Discussion Type
- discussion topichigher dimensional WZW theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 28th 2011

I am starting something at higher dimensional WZW theory

- Discussion Type
- discussion topicfalse vacuum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 27th 2011

stub for false vacuum

- Discussion Type
- discussion topiccohomology of local net of observables
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Oct 26th 2011

created cohomology of local net of observables. I have included a brief Idea-section but mainly this is, for the moment, to record references.

- Discussion Type
- discussion topiccombinatorial Hopf algebras
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 25th 2011

New entry combinatorial Hopf algebra. Reference additions or updates in Hopf algebra, BV formalism, Hall algebra, graph homology, Marcelo Aguiar, renormalization.

- Discussion Type
- discussion topichorizon
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by TobyBartels
- Last Active Oct 25th 2011

Have a look at horizon.

- Discussion Type
- discussion topicClassifying topos of a localic groupoid
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Oct 23rd 2011

- I added the following remark to classifying topos of a localic groupoid.

It would be nice if somebody more competent in this area expanded it.

The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in

Ieke Moerdijk, The classifying topos of a continuous groupoid II,

Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168.

- Discussion Type
- discussion topicWZW-type superstring field theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 21st 2011

stub for WZW-type superstring field theory

- Discussion Type
- discussion topicSeiberg duality
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Oct 21st 2011

stub for Seiberg duality motivated from this discussion at “Theoretical Physics Stack Exchange” (is this now publically visible?)

- Discussion Type
- discussion topic0-morphism
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by TobyBartels
- Last Active Oct 19th 2011

felt the need for 0-morphism

- Discussion Type
- discussion topicdirectional derivative
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 19th 2011

New entry directional derivative, redirecting also Gâteaux derivative. Much of the material is adapted from Fréchet space (which also calls for this entry). Somebody should write more on the (possibly infinite-dimensional) manifold case.

- Discussion Type
- discussion topicStephan's edits
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by zskoda
- Last Active Oct 19th 2011

Somebody signing as “Stephan” has made half a dozen or so edits lately. Does anyone know who this might be? Because I would like to suggest to him to announce his changes here.

Mostly they were very useful corrections. But at nice category of spaces and at groupoid object in an (infinity,1)-category I felt that the comment added there by him was in need of a bit of rephrasing. Nothing serious, but I’d like to know who he is to sort this out.

He also, I think, created a new entry titled Principal bundles, groupoids and connections

- Discussion Type
- discussion topichomotopical algebra
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by jim_stasheff
- Last Active Oct 19th 2011

I have cross-linked the two entries homotopical algebra and higher algebra.

At homotopical algebra I moved the text that had existed there into a subsection “History”, because that’s what it is about, right? I added a section “Idea” but so far only included a link to higher algebra there. We could maybe merge the two entries.

- Discussion Type
- discussion topicconnection on a bundle
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Oct 18th 2011

added to connection on a bundle

a Definition (nPOV-flavor, of course)

a Properties-section with statement and proof of the fact that every bundle does admit a connection.

- Discussion Type
- discussion topictable of orthogonal groups and related
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by zskoda
- Last Active Oct 17th 2011

started something called table of orthogonal groups and related and included it into the relevant entries

- Discussion Type
- discussion topicdg-manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 17th 2011

stub for dg-manifold and dg-scheme

- Discussion Type
- discussion topictable of branes
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 14th 2011

I have started creating a

*table of branes and their worldvolume theories*. So far it looks as followsI have created all the missing entries to complete this and have included the table in all relevant entries.

- Discussion Type
- discussion topic7d supergravity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 14th 2011

I am starting an entry 7-dimensional supergravity in order to collect some references that I need

- Discussion Type
- discussion topicAdS7 / CFT6
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 13th 2011

I have been adding to AdS/CFT in the section

*AdS7 / CFT6*a (of course incomplete) list of available evidence for what is going on.This is triggered by the fact that we have a proposal for a precise formalization of the effective 7d theory.

- Discussion Type
- discussion topicconfiguration space
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by zskoda
- Last Active Oct 13th 2011

stub for configuration space with $\infty$-topos theoretic definition. See also phase space

- Discussion Type
- discussion topicn-poset
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 13th 2011

I have added an explanatory paragraph to n-poset in reply to this MO question.

Also, at poset itself I have added a word (“hence”) to indicate that if something is a category with at most one morphism between any ordered pair of objects, then it is already implied that if there are two morphisms back and forth between two objects, then these are equal.

- Discussion Type
- discussion topicepi/mono factorization system
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by TobyBartels
- Last Active Oct 12th 2011

I have split off an entry epi/mono factorization system in order to better be able to amplify the higher pattern that this sits in

- Discussion Type
- discussion topichomotopy image
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 10th 2011

added to homotopy image a brief remark in a new subsection on how this is given by the n-connected/n-truncated factorization system for $n = -1$.

- Discussion Type
- discussion topicn-connected/n-truncated factorization system
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 10th 2011

at n-connected/n-truncated factorization system I have created an Examples-section with a brief indication of what this factorization “means” for low values of $n$ (from $n= -2$ to $n = 0$).

- Discussion Type
- discussion topicRevamping [[measurable space]].
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 9th 2011

I plan to redo measurable space, and the outline of the plan is now at the bottom.

For the nonce, I’ve moved some material to a new article sigma-algebra, and some of that thence to the previous stub Borel subset.

- Discussion Type
- discussion topicnew entries surrounding well-quasi-orders
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Todd_Trimble
- Last Active Oct 6th 2011

Stubby beginnings of articles on well-quasi-order, antichain, and graph minor. Some minor mention (ha ha) of the Robertson-Seymour theorem. Please feel free to add more.

- Discussion Type
- discussion topicHamiltonian vector fields on symplectic oo-groupoids
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 6th 2011

new section at symplectic infinity-groupoid on Hamiltonian vector fields on symplectic oo-groupoids.

- Discussion Type
- discussion topicgeometric Langlands program
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Oct 5th 2011

I have significantly extended the list of references at geometric Langlands program. Langlands’ is here in the role of adjective and geometric Langlangds is an informal abbreviation. I have changed the former name geometric Langlands to geometric Langlands program but due cache bug now two pages seem to exist in parallel.

- Discussion Type
- discussion topiccollective field theory, large N limit
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 4th 2011

On Sep 29, I added the new stubs collective field theory and large N limit. My interest is in the question I just posed at theoreticalPhysics.stackexchange systematic-approach-to-deriving-equations-of-collective-field-theory-to-any-order.

In my opinion/wish, I should have been better prepared to ask that question (in depth reading of some of the key references are on my todo list), but I posted the question a bit earlier than ready for a better documented question, as anyway there is a need of constructive kick-off of the activity at thPh.stExch.

- Discussion Type
- discussion topicThe algebra of idempotents
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Oct 3rd 2011

I wrote about the boolean algebra of idempotents in a commutative ring. There’s also stuff in there about projection operators (that page doesn’t exist).

- Discussion Type
- discussion topicUpdated reference in Tall-Wraith monoid
- Category Latest Changes
- Started by Andrew Stacey
- Comments 5
- Last comment by Andrew Stacey
- Last Active Sep 27th 2011

Updated the reference to "The Hunting of the Hopf Ring" since it's now appeared in print.

- Discussion Type
- discussion topicLCC pretopos
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by Todd_Trimble
- Last Active Sep 21st 2011

I added a comment to the end of the discussion at predicative mathematics to the effect that free small-colimit completions of toposes are examples of locally cartesian closed pretoposes that are generally not toposes.

- Discussion Type
- discussion topiccurve
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Sep 20th 2011

I added the notion of a regular curve to curve. In differential geometry, for most purposes only regular curves are useful: the parametrized smooth curves with never vanishing velocity. Smooth curves as smooth maps from the interval are not of much use without the regularity condition: their image may be far from smooth, with e.g. cusps and clustered sequences of self/intersections.

- Discussion Type
- discussion topicHigher local Kan extensions
- Category Latest Changes
- Started by fpaugam
- Comments 5
- Last comment by fpaugam
- Last Active Sep 20th 2011

- Do you have some ideas on how to define a general/higher notion of local Kan extension in an n-category, that gives back the usual notion in a 2-category? I am talking of local kan extension Lan_F G, with F and G two morphisms, that is given by a 2-cell with F and G on the boundary that is universal among such 2-cells.

I would define it using as in the nlab page, the corepresentation of the functor Hom(G,F^*_) but this does not make sense in a weak n-category. I don't want of a Lurie type kan extension given by adjoint to F^*. Want something weaker.

One could also use simply truncation to a 2-category, but is there something finer than that?

The applications i have in mind are related to higher doctrines and theories, derived algebras and their universal properties.

Is there in the litterature something finer than that and useful?

- Discussion Type
- discussion topicLie 2-group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 20th 2011

added stuff to Lie 2-group: more in the Idea-section, more examples, some constructions, plenty of references.

- Discussion Type
- discussion topicInfinite descent
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Sep 19th 2011

I rephrased the classical alternative formulations at well-founded relation to define relations with no infinite descent.

- Discussion Type
- discussion topicintroduction to oo-Chern-Weil theory
- Category Latest Changes
- Started by Urs
- Comments 43
- Last comment by domenico_fiorenza
- Last Active Sep 19th 2011

I am working on further bringing the entry

infinity-Chern-Weil theory introduction

into shape. Now I have spent a bit of time on the (new) subsection that exposes just the standard notion of principal bundles, but in the kind of language (Lie groupoids, anafunctors, etc) that eventually leads over to the description of smooth principal oo-bundles.

I want to ask beta-testers to check this out, and let me know just how

*dreadful*this still is ! ;-) The section I mean is at

- Discussion Type
- discussion topicLoops with singularities
- Category Latest Changes
- Started by Andrew Stacey
- Comments 10
- Last comment by Andrew Stacey
- Last Active Sep 19th 2011

I found an interesting question on MO (here) and merrily set out to answer it. The answer got a bit long, so I thought I’d put it here instead. Since I wrote it in LaTeX with the intention of converting it to a suitable format for MO, it was simplicity itself to convert it instead to something suitable for the nLab.

The style is perhaps not quite right for the nLab, but I can polish that. As I said, the original intention was to post it there so I started writing it with that in mind. I’ll polish it up and add in more links in due course.

The page is at: on the manifold structure of singular loops, though I’m not sure that that’s an appropriate title! At the very least, it ought to have a subtitle: “or the lack of it”.

- Discussion Type
- discussion topicmodel structure on cosimplicial spaces
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 19th 2011

created model structure on cosimplicial simplicial sets, mainly in order to record two references.

- Discussion Type
- discussion topicanomaly line bundle for self-dual higher gauge theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 17th 2011

added today’s article by Monnier to the references at quantum anomaly and self-dual higher gauge theory

- Discussion Type
- discussion topicKilling tensor and Killing-Yano tensor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 17th 2011

quick stubs for Killing tensor and Killing-Yano tensor in reply to a question here

- Discussion Type
- discussion topicPaths and loops
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 17th 2011

- Discussion Type
- discussion topicgeometric quantization of symplectic infinity-groupoids
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by zskoda
- Last Active Sep 14th 2011

at geometric quantization of symplectic infinity-groupoids (which currently still redirects to symplectic infinity-groupoid) I am beginning to add some genuine substance. So far:

an Idea-section

and a section Prequantum circle (n+1)-bundle with the general abstract definition and the beginning of some examples

- Discussion Type
- discussion topicFrobenius map
- Category Latest Changes
- Started by hilbertthm90
- Comments 8
- Last comment by zskoda
- Last Active Sep 14th 2011

I created Frobenius map, since I had linked to it in several places.

- Discussion Type
- discussion topicChern-Simons element
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Sep 11th 2011

added to Chern-Simons element in the section Standard Chern-Simons form a detailed pedestrian proof of how the standard Chern-Simons 3-form is reproduced from this machinery.

- Discussion Type
- discussion topicWell-orders need not be linear.
- Category Latest Changes
- Started by TobyBartels
- Comments 11
- Last comment by TobyBartels
- Last Active Sep 11th 2011

On well-order and elsewhere, I’ve implied that a well-order (a well-founded, extensional, transitive relation) must be connected (and thus a linear order). But this is not correct; or at least I can’t prove it, and I’ve read a few places claiming that well-orders need not be linear. So I fixed well-order, although the claim may still be on the Lab somewhere else.

Of course, all of this is in the context of constructive mathematics; with excluded middle, the claim is actually true. I also rewrote the discussion of classical alternatives at well-order to show more popular equivalents.

- Discussion Type
- discussion topicF-categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Sep 11th 2011

Created F-category and rigged limit.

- Discussion Type
- discussion topicfactorization structures
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Sep 10th 2011

Created factorization structure for sinks, and added remarks to Grothendieck fibration, M-complete category, and topological concrete category about constructing them. Largely I wanted to record the proof of the theorem that in an factorization structure for sinks $(E,M)$, the class $M$ necessarily consists of monomorphisms. It’s a nice generalization of Freyd’s theorem about complete small categories, which has more of the feel of a useful theorem than of a no-go theorem like Freyd’s.

At first I thought that the lifting of factorization structures described at topological concrete category would work for solid functors too, but then I couldn’t see how to do it. Does anyone?

- Discussion Type
- discussion topicKZ equations
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 10th 2011

[Jim Stasheff means to post the following message here to the $n$Forum, but accidentally (I think) posted it here instead (I guess because that is the forum that comes up when one googles for “nForum”)]:

Solutions of the KZ equations are usually given in terms of assymptotic behavior in certain regions. Since the region on which the eqns are defined has a nice compactication, what is the obstruction to extending the solutions to the compactification?

- Discussion Type
- discussion topicsymmetric space, quandle
- Category Latest Changes
- Started by John Baez
- Comments 11
- Last comment by zskoda
- Last Active Sep 9th 2011

I started a stub

and added a bit of overlapping material to quandle. I would like to talk about Lie and Jordan triple systems, but I need this introductory material first.

- Discussion Type
- discussion topicM-complete categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by TobyBartels
- Last Active Sep 7th 2011

Created M-complete category.

- Discussion Type
- discussion topiccenter and outer automorphisms of infinity-groups
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 7th 2011

created

(to go with the discussion at ∞-gerbe).

There are other notions of “center” of $\infty$-groups. But this is one of them.

- Discussion Type
- discussion topicSimulations
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by TobyBartels
- Last Active Sep 7th 2011

Inspired by Todd’s work at well-founded relation, I’ve written simulation.

- Discussion Type
- discussion topiclax morphism
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by TobyBartels
- Last Active Sep 7th 2011

I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.