Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical 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 differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory 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 limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural 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 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicHodge theory
- Category Latest Changes
- Started by Kevin Lin
- Comments 8
- Last comment by Eric
- Last Active Jun 26th 2010

I added a bit of stuff to the Hodge theory page and started a Noncommutative Hodge theory page. I hope to expand on these pages soon.

- Discussion Type
- discussion topicreflective subcategory
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by Todd_Trimble
- Last Active Jun 26th 2010

I did a little bit of rewriting and cleaning up at reflective subcategory, in an effort to make things clearer for the neophyte. Part of the cleaning-up was to remove a query initiated by Zoran under the section Characterizations (I rewrote a bit to make the question vanish altogether).

There’s another query of Zoran at the bottom which I think was answered by Mike, but let me ask before removing it.

- Discussion Type
- discussion topicQuery at Omega-group
- Category Latest Changes
- Started by John Baez
- Comments 3
- Last comment by zskoda
- Last Active Jun 26th 2010

- Aleks Kleyn emailed me saying he would like a reference or two to work on Ω-groups, so maybe someone can help him out. I put a query on the relevant page.

- Discussion Type
- discussion topicproblem with editing
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Jun 25th 2010

I can not edit Croatian black hole school, the edit button redirects me to the Homepage.

- Discussion Type
- discussion topicasymptotic isometry
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jun 25th 2010

- Discussion Type
- discussion topicsplit sequence
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Andrew Stacey
- Last Active Jun 25th 2010

Charles Siegel created split sequence

- Discussion Type
- discussion topicTCFT
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by Urs
- Last Active Jun 25th 2010

started TCFT

eventually we should better reflect at the relevant nLab entries that Costello’s classification result of 2d TCFTs – proving the corresponding conjecture by Kontsevich – was the precurser of the proof of the full cobordism hypothesis: Hopkins and Lurie tried to generalize Costello’s proof (they had not actually heard of Baez-Dolan back then).

- Discussion Type
- discussion topicC*-algebras
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jun 25th 2010

I made some edits to C-star-algebra and representation of a C-star-algebra.

- Discussion Type
- discussion topicNotation query from separation axioms
- Category Latest Changes
- Started by Andrew Stacey
- Comments 23
- Last comment by TobyBartels
- Last Active Jun 24th 2010

Just got the following query from Harald Hanche-Olsen about the page separation axioms. As I’ve never seen that notation before either (but agree with Harald’s comments in both parts), I’m forwarding it here so that the person who first adopted it (Toby?) or others can chip in.

I hadn’t seen the notation $\stackrel\circ\ni$ for a neighbourhood before, but it looks like a reasonable notation that I might want to adapt. BUT it seems more appropriate for a neighbourhood of a point rather than a neighbourhood of a set. Wouldn’t $\stackrel\circ\supset$ or $\stackrel\circ\supseteq$ be more appropriate for that case? What is the rationale for the usage on that page?

- Discussion Type
- discussion topicorthogonality
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Harry Gindi
- Last Active Jun 24th 2010

edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.

And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?

I think the entry-situation here deserves to be further harmonized.

- Discussion Type
- discussion topic[orthogonal subcategory problem]
- Category Latest Changes
- Started by Harry Gindi
- Comments 5
- Last comment by Urs
- Last Active Jun 24th 2010

Over at orthogonal subcategory problem, it’s not clear to me whether or not the “objects orthogonal to $\Sigma$” should be morphisms orthogonal to $\Sigma$, or if it should mean objects of $X$ of $C$ such that $X\to *$ is orthogonal to $\Sigma$ (where $*$ denotes the terminal object). (Hell, it could even mean objects that are the source of a map orthogonal to $\Sigma$). I was in the process of changing stuff to fit the first interpretation, but I rolled it back and decided to ask here.

If it should in fact be the second (or third) definition, I would definitely suggest changing the notation $\Sigma^\perp$, which is extremely misleading, since that is the standard notation for the first notion.

- Discussion Type
- discussion topicspin-statistics theorem
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Ian_Durham
- Last Active Jun 23rd 2010

stub for spin-statistics theorem. Just recording a first few references so far.

- Discussion Type
- discussion topicCircle Lie n-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 23rd 2010

added to Lie infinity-groupoid a section on Circle Lie n-groupoids, i.e. those of the form $\mathbf{B}^n U(1)$, and their relation to Cech- , Deligne-, and de Rham cohomology.

- Discussion Type
- discussion topicG-delta subset
- Category Latest Changes
- Started by Andrew Stacey
- Comments 2
- Last comment by TobyBartels
- Last Active Jun 23rd 2010

Created G-delta subset of a topological space, and its snappier redirect: G-delta. (Technically, these ought to be $G_\delta$, I guess, which de-mathemalises to Gδ but I preferred spelling the delta out in full, does that sound okay?).

This is mainly to record a result about completely regular spaces in which every point is a G-delta subset which relates to the result I put up on sequentially compact space which in turn is related to the question of when the curvaceous topology and functional topology of a Froelicher space agree.

- Discussion Type
- discussion topicPoincare conjecture
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 22nd 2010

since it was demanded at the “counterexamples”-page, I created 3-manifold. This made me create Poincare conjecture.

I find it striking that Hamilton’s Ricci flow program and Perelman’s proof by adding the dilaton hasn’t found more resonance in the String theory community. After all, this shows a deep fact about the renormalization group flow of non-critical strings on 3-dimensional targets with gravity and dilaton background.

I once chatted with Huisken and indicated that this suggests that there is a more general interesting mathematical problem where also the Kalb-Ramond field background is taken into account. I remember him being interested, but haven’t heard that anyone in this area has extended Perelman’s method to the full massles string background content. Has anyone?

- Discussion Type
- discussion topicfat simplex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 22nd 2010

felt like creating fat simplex in parallel to Bousfield-Kan map

- Discussion Type
- discussion topicCounterexamples in Algebra
- Category Latest Changes
- Started by Andrew Stacey
- Comments 17
- Last comment by Urs
- Last Active Jun 22nd 2010

counterexamples in algebra inspired (and largely copied from) this MO question since MO is a daft place to put that stuff and a page on the nLab seems better. (A properly indexed database would be even better, but I don’t feel like setting such up and don’t know of the existence of such a system)

- Discussion Type
- discussion topicAdded a result on sequentially compact spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 5
- Last comment by Andrew Stacey
- Last Active Jun 22nd 2010

I’ve added a result to the list at sequentially compact space which is an analogue of the more well-known one about compact Hausdorff spaces. This also relates to this MO question.

- Discussion Type
- discussion topicC* algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Jun 21st 2010

added to C-star algebra a stub section on the dagger-categorical formulation

- Discussion Type
- discussion topicBell's theorem
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Jun 21st 2010

Cleaned up Bell’s theorem a bit in my ongoing effort to better organize and clean up the quantum mechanics entries.

- Discussion Type
- discussion topicFell's theorem
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 4
- Last comment by Urs
- Last Active Jun 21st 2010

As a small step towards more information about representations of operator algebras and their physical interpretation in AQFT, I extraced states from operator algebras and added Fell’s theorem. This is a theorem that is often cited in the literature, but most times not with any specific name (often with no reference, either). But I think it is both justified and usefule to call it Fell’s theorem :-)

- Discussion Type
- discussion topicTopological locales
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by Mike Shulman
- Last Active Jun 19th 2010

I got tired of making unmatched links to topological locale (aka spatial locale, or locale with enough points), so I wrote a stub.

- Discussion Type
- discussion topicenriched category theory contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2010

started floating toc enriched category theory contents and added it to relevant entries

- Discussion Type
- discussion topiccoskeleton and truncation
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 18th 2010

I am trying to remove the erroneous shifts in degree by $\pm 1$ that inevitably I have been making at simplicial skeleton and maybe at truncated.

So a Kan complex is the nerve of an $n$-groupoid iff it is $(n+1)$-coskeletal, I hope ;-)

At truncated in the examples-section i want to be claiming that the truncation adjunction in a general (oo,1)-topos is in the case of $\infty$Grpd the $(tr_{n+1} \dashv cosk_{n+1})$-adjunction on Kan complexes. But I should be saying this better.

- Discussion Type
- discussion topicMass
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Ian_Durham
- Last Active Jun 17th 2010

The mass of a physical system is its intrinsic energy.

I expect that Zoran will object to some of what I have written there (if not already to my one-sentence definition above), but since I cannot predict how, I look forward to his comments.

- Discussion Type
- discussion topiccentipede pictures
- Category Latest Changes
- Started by zskoda
- Comments 65
- Last comment by zskoda
- Last Active Jun 17th 2010

John Baez has erased our query complaining about disgusting picture at quasigroup, and left the picture. I like the theory of quasigroups but do not like to visit and contribute to sites dominated by strange will to decorate with self-proclaimed humour which is in fact tasteless.

- Discussion Type
- discussion topicCartSpace
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by zskoda
- Last Active Jun 17th 2010

added to CartSp a section that lists lots of notions of (generalized) geometry modeled on this category.

- Discussion Type
- discussion topiccategorical homotopy groups in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 17th 2010

expanded categorical homotopy groups in an (infinity,1)-topos

added more details to the definition of the homotopy sheaves;

added a section on how the Joyal-Jardine homotopy sheaves of simplicial presheaves are a model for that.

- Discussion Type
- discussion topichomotopy groups in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 17th 2010

I keep working on the entry now titled homotopy groups in an (infinity,1)-topos (used to be "homotopy groups of an oo-stack").

The following subsections I added

Geometric Pi_oo of the terminal object in a locally connected oo-topos

Geometric Pi_0 of a genral object in a locally connected topos

Examples (meaning: general examples, neither purely categorical nor geometrical, currently a discussion of simple examples that distinguish the two notions)

This all needs more work. But I'll stop for a moment and instead start now an entry on locally n-connected (oo,1)-toposes, which I need for further discussion here.

- Discussion Type
- discussion topic[[Connes fusion]]
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 5
- Last comment by domenico_fiorenza
- Last Active Jun 16th 2010

continued from here

my proposal:

Connes fusion is used to define fusion of positive energy representations of the loop group $\mathcal{L}SU(N)$ in * Antony Wassermann, Operator algebras and conformal field theory III (arXiv) and to define elliptic cohomology in * Stephan Stolz and Peter Teichner, What is an elliptic object? (link)

and removing the query box.

- Discussion Type
- discussion topicsubscheme of an Abelian category
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Jun 15th 2010

New entry to support the discussion with Urs about infinitesimally thickened topos.

- Discussion Type
- discussion topicconormal bundle
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jun 15th 2010

- Discussion Type
- discussion topicinfinitesimally thickened topos
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by zskoda
- Last Active Jun 15th 2010

Some of you may remember that a while ago I had started wondering how one could characterize geometric morphisms of toposes $E \to F$ that would exhibit $E$ as an “infinitesimal thickening” of $F$.

Instead of coming to a defnite conclusion on this one, I worked with a concrete example that should be an example of this situation: that of the Gorthendieck toposes on the sites CartSp and ThCartSp of cartesian spaces and infinitesimally thickened cartesian spaces.

But now I went through my proofs for that situation and tried to extract which abstract properties of these sites they actually depend on. Unless I am mixed up, it seems to me now that the essential property is $CartSp$ is a

$CartSp \stackrel{\leftarrow }{\hookrightarrow} ThCartSp$*coreflective subcategory*of $ThCartSp$ and that in the respective adjunctionbuth functors preserve covers.

So maybe it makes sense to take this as a definition: a geometric morphism of Grothendieck toposes is an infinitesimal thickening if it comes from such a coreflective embedding of sites.

Details of this, with more comments on the meaning of it all and detailed proofs, I have now typed into my page on path oo-functors in the section Infinitesimal path oo-groupoids.

- Discussion Type
- discussion topicdifferential bimodule
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 15th 2010

added to differential bimodule the basic example, thanks to Zoran

- Discussion Type
- discussion topicConjunctions
- Category Latest Changes
- Started by TobyBartels
- Comments 21
- Last comment by Todd_Trimble
- Last Active Jun 15th 2010

I added a disambiguation note to conjunction, since most of the links to that page actually wanted something else. Then I changed those links to something else: logical conjunction (not yet extant).

An Internet and dictionary search suggests that there is no analogous danger for disjunction (also not yet extant).

- Discussion Type
- discussion topictwo-sided bar construction
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by Mike Shulman
- Last Active Jun 14th 2010

Wrote two-sided bar construction. There is a lot to add, but I added a query box under the subsection “Delooping machines” which I’m hoping someone like Mike could answer.

- Discussion Type
- discussion topicHilbert space QM
- Category Latest Changes
- Started by Ian_Durham
- Comments 1
- Last comment by Ian_Durham
- Last Active Jun 12th 2010

Moved the subsection on Hilbert space quantum mechanics from quantum information to quantum mechanics as had been suggested by someone in a query box.

- Discussion Type
- discussion topicmate
- Category Latest Changes
- Started by FinnLawler
- Comments 12
- Last comment by FinnLawler
- Last Active Jun 11th 2010

I’ve created mate, but I can’t get my nice fancy diagrams to display – I just get the source. Help! What have I done wrong?

- Discussion Type
- discussion topicUnbounded posets
- Category Latest Changes
- Started by TobyBartels
- Comments 16
- Last comment by Tim_van_Beek
- Last Active Jun 11th 2010

Tim van Beek has written about unbounded posets at partial order.

Where is this used?

- Discussion Type
- discussion topic[local isomorphism]
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by DavidRoberts
- Last Active Jun 11th 2010

In another thread I came up with a definition of a local isomorphism in a site, working from the definition of a local homeomorphism/diffeomorphism in Top/Diff respectively (with the open cover pretopology in both cases). Then I find that there is a page local isomorphism talking about maps in presheaf categories: such a map is a local isomorphism if becomes an isomorphism on applying the sheafification functor $PSh(S) \to Sh(S,J)$. To quote my definition again

**Definition:**Let (C,J) be a site (J a pretopology). A map $f:a \to b$ is a J-local isomorphism if there are covering families $(v_i \to b)$ and $(u_j \to a)$ such that for each $u_j$ the restriction $f|u_j$ is an isomorphism onto some $v_i$.I don’t claim, in the time I have available, to understand the implications of the definition at local isomorphism. I just wonder how it relates to concrete notions like local homeomorphisms (let us work with Top and open covers as covering families). Is a local homeomorphism, after applying Yoneda, a local isomorphism? Does a local isomorphism in the image of Yoneda come from a local homeomorphism? I suspect the answer is yes. Now for the biggie: can a local isomorphism be characterised in terms as basic as my definition as quoted? With my definition one avoids dealing with functor categories (and so size issues, to some extent: $[Top^{op},Set]$ is very big), so if they are equivalent, I’d like to put this somewhere.

Obviously we can take the site in my definition to be a presheaf category with the canonical pretopology or something, and potentially recover the definition at local isomorphism, but for the ease of connecting with geometric ideas, I prefer something simpler.

Any thoughts?

- Discussion Type
- discussion topicsmall and large sites
- Category Latest Changes
- Started by DavidRoberts
- Comments 55
- Last comment by zskoda
- Last Active Jun 10th 2010

Created small site and large site of an object in a site, as a spin off from discussion around petit topos. The latter is so named because large site is taken for sites that happen to be large. The content of this page, however, looks as though it could go somewhere discussing sheaves.

- Discussion Type
- discussion topicHisham Sati
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by Urs
- Last Active Jun 10th 2010

Urs has erased the sentence explanining the purpose of the entry. Why ??

"In fact not only that it is a good survey but it has a nice bibliography. The main plan of this entry is to build a hyperlinked bibliography of the above article!"

- Discussion Type
- discussion topicSmooth paths
- Category Latest Changes
- Started by Andrew Stacey
- Comments 18
- Last comment by Andrew Stacey
- Last Active Jun 9th 2010

Started thinking about smooth paths.

(Incidentally, David, do you want query boxes added to your web? And would you like to change the CSS for off-web links from those boxes to some nice colour?)

- Discussion Type
- discussion topicBaire property
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jun 8th 2010

I felt the need to write down what it means for a subspace to have the Baire property, so I did.

- Discussion Type
- discussion topicclosed monoidal structure on an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 8th 2010

A discussion of the cartesian closed monoidal structure on an (oo,1)-topos is currently missing on the nLab.

I started making a first step in the direction of including it:

at model structure on simplicial presheaves I added a section Closed monoidal structure with a pointer to Toen’s lectures (where the following is an exercise) and a statement and proof of how $[C^{op},sSet]_{proj}$ is a monoidal model category by the Cartesian product.

as a lemma for that I added to Quillen bifunctor the statement that on cofib generated model cats a Quillen bifunctor property is checked already on generating cofibrations (here).

More later…

- Discussion Type
- discussion topicYoneda-related stuff
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Jun 7th 2010

Based on recent discussions here primarily with Harry and Urs, I added a proof at co-Yoneda lemma in terms of extranaturality, and some didactic material over at adjunction bridging hom-functors to units/counits, involving some but hopefully not too much overlap with related material Urs recently added at adjoint functor. Still need to work in some links.

- Discussion Type
- discussion topicparacategories
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by Mike Shulman
- Last Active Jun 7th 2010

Created paracategory and Kleene equality.

- Discussion Type
- discussion topic[cylinder on a presheaf]
- Category Latest Changes
- Started by Harry Gindi
- Comments 12
- Last comment by Harry Gindi
- Last Active Jun 7th 2010

I created cylinder on a presheaf and will fill it in more as I read through Ast308. I plan on adding more stuff as I get to it (things about test categories and localisers, etc.).

This is similar but not the same as cylinder object, since even though it is specialized to presheaf categories, we don’t require any notion of a weak equivalence a priori.

- Discussion Type
- discussion topicSurface diagrams
- Category Latest Changes
- Started by Todd_Trimble
- Comments 44
- Last comment by Mike Shulman
- Last Active Jun 6th 2010

I have quietly submitted the beginning of an article on "surface diagrams" on my web. There is still quite a lot left to write up, and it needs to be formatted more prettily, but I thought I'd throw what I have (so far) out there.

- Discussion Type
- discussion topicpre-Lie algebras
- Category Latest Changes
- Started by John Baez
- Comments 1
- Last comment by John Baez
- Last Active Jun 6th 2010

- I have started an entry on pre-Lie algebras, which are much more interesting than you might think at first. My friend Bill Schmitt, the combinatorist, is visiting and telling me amazing things about combinatorics and operads.... this is a little bit of the story.

- Discussion Type
- discussion topictopological stack
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 6th 2010

Dave Carchedi has been editing/expanding topological stack

- Discussion Type
- discussion topic[Kan extension]
- Category Latest Changes
- Started by Harry Gindi
- Comments 2
- Last comment by Mike Shulman
- Last Active Jun 6th 2010

I moved the characterization of pointwise kan extensions as those preserved by representable functors to the top (of the section on pointwise kan extensions) and made it the definition (since there was no unified definition before). This is for aesthetic reasons. Since being pointwise is a

*property*, I like that this property has a definition independent of the computational model we’re using.Are there size issues that I might be glossing over?

- Discussion Type
- discussion topicAC00
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Jun 5th 2010

I added a paragraph about $AC_{00}$ to countable choice

- Discussion Type
- discussion topicGrothendieck construction
- Category Latest Changes
- Started by Mike Shulman
- Comments 11
- Last comment by Urs
- Last Active Jun 3rd 2010

I think the definition of the Grothendieck construction was wrong. The explicit definition was right, but the description in terms of a generalized universal bundle didn’t work out to that, if by “the category of pointed categories” was meant for the functors to preserve the points, which is the usual meaning of a category of pointed objects. I corrected this by using the lax slice. Since while I was writing it I got confused with all the op’s, I decided that the reader might have similar trouble, so I changed it to do the covariant version first and then the contravariant.

- Discussion Type
- discussion topicpetit topos / gros topos
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Harry Gindi
- Last Active Jun 3rd 2010

I expanded the Examples-section at petit topos and included a reference to Lawvere’s “Axiomatic cohesion”, which contains some discussion of some aspects of a characterization of “gros” vs “petit” (which I wouldn’t have noticed were it not for a talk by Peter Johnstone).

I am thinking that it should be possible to give more and more formal discussion here, using Lawvere’s article and potentially other articles. But that’s it from me for the time being.

- Discussion Type
- discussion topic[[limit]]
- Category Latest Changes
- Started by Harry Gindi
- Comments 2
- Last comment by Mike Shulman
- Last Active Jun 2nd 2010

Swapped the order of the propositions that small limits commute with small limits and that limits commute with right adjoints, which allowed me to give a proof that small limits commute with small limits by citing the result on right adjoints and the characterization of the limit as right adjoint to the constant diagram functor.

- Discussion Type
- discussion topicdependent choice
- Category Latest Changes
- Started by Todd_Trimble
- Comments 26
- Last comment by TobyBartels
- Last Active Jun 1st 2010

Started the article dependent choice, and did some editing at COSHEP to make clearer to myself the argument that COSHEP + (1 is projective) implies dependent choice. It’s not clear to me that the projectivity of 1 is removable in that argument; maybe it is.

- Discussion Type
- discussion topicFamilies of sets
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jun 1st 2010

Started a stub at family of sets. This should also explain concepts like a family of subsets of a given set or a family of groups. And how to formalise them all in material and structural set theories, predicative foundations, internally in indexed categories, etc.

- Discussion Type
- discussion topicquantum field theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Ian_Durham
- Last Active Jun 1st 2010

wrote an Idea-section at quantum field theory

- Discussion Type
- discussion topicHausdorff dimension
- Category Latest Changes
- Started by TobyBartels
- Comments 10
- Last comment by Eric
- Last Active Jun 1st 2010

An anonymous coward put something blank (or possibly some spam that somebody else blanked within half an hour) at Hausdorff dimension, so I put in a stub.