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- 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 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 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 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 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.

- Discussion Type
- discussion topicdifferentiable Lie group cohomology as intrinsic (oo,1)-topos cohomology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active May 31st 2010

I moved the proof of the claim that the Segal-Brylinski “differetiable Lie group cohomology” is that computed in the (oo,1)-topos of oo-Lie groupoids from the entry group cohomology to the entry Lie infinity-groupoid and expanded the details of the proof considerably.

See this new section.

Towards the end I could expand still a bit more, but I am not allowed to work anymore today… :-)

- Discussion Type
- discussion topicStacks and queues
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active May 30th 2010

I’ve added a bit about these to free monoid. (These are the computer scientists’ stacks, not the geometers’ stacks!) There is a query about queues too; I’ve forgotten something and can’t reconstruct it.

- Discussion Type
- discussion topicname change
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Todd_Trimble
- Last Active May 30th 2010

Changed a page title from topological topos to Johnstone’s topological topos. Urs said I should call for help when making a name change, so that someone can clear the cache to get the change to propagate properly.

- Discussion Type
- discussion topicbasis
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by zskoda
- Last Active May 29th 2010

started a disambiguation page basis

- Discussion Type
- discussion topicnonabelian+homological+algebra
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active May 29th 2010

I just started nonabelian homological algebra.

- Discussion Type
- discussion topiccoverages and localizations
- Category Latest Changes
- Started by zskoda
- Comments 26
- Last comment by zskoda
- Last Active May 29th 2010

Regarding that the nlabizens have discussed so much various generalizations of Grothendieck topology, maybe somebody knows which terminology is convenient for the setup of covers of abelian categories by finite conservative families of flat localizations functors, or more generally by finite conservative families of flat (additive) functors. Namely the localizations functors do not mutually commute so the descent data are more complicated but if you produce the comonad from a cover then the descent data are nothing but the comodules over the comonad on the product of the categories which cover. In noncommutative geometry we often deal with stacks in this generalization of topology and use ad hoc language, say for cocycles, but the thing is essentially very simple and the language barier should be overcome. There are more general and ore elaborate theories of nc stacks, but this picture is the simplest possible.

- Discussion Type
- discussion topiccrystalline cohomology
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active May 28th 2010

stub for crystalline cohomology

There are notes by Jacob Lurie on crystals, but I forget where to find them. Does anyone have the link?

- Discussion Type
- discussion topicDiagram of locally convex TVS properties
- Category Latest Changes
- Started by Andrew Stacey
- Comments 6
- Last comment by Andrew Stacey
- Last Active May 27th 2010

I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram. So I’ve gone for a more modest aim, that of trying to convey a little more information than the original diagram.

Unfortunately, the nLab isn’t displaying the current diagram, though the original one displays just fine and on my own instiki installation then it also displays just fine so I’m not sure what’s going on there. Until I figure that out, you can see it here. The source code is in the nLab: second lctvs diagram dot source.

A little explanation of the design:

- Abbreviate all the nodes to make the diagram more compact (with a key by the side, and tooltips to display the proper title).
- Added some properties: LF spaces, LB spaces, Ptak spaces, $B_r$ spaces
- Taken out some properties: I took out those that seemed “merely” topological in flavour: paracompactness, separable, normal. I’m pondering taking out completeness and sequential completeness as well.
- Tried to classify the different properties. I picked three main categories: Size, Completeness, Duality. By “Size”, I mean “How close to a Banach space?”.

(It seems that Instiki’s SVG support has … temporarily … broken. I’ll email Jacques.)

- Discussion Type
- discussion topicbasis for a topology
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by zskoda
- Last Active May 27th 2010

created basis for a topology and linked to it with comments from coverage and, of course, Grothendieck topology

- Discussion Type
- discussion topictensoring over ooGrpd
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 27th 2010

added a still somewhat stubby section on tensoring over ooGrpd to limits in a quasi-category

- Discussion Type
- discussion topic(oo,1)-category of (oo,1)-sheaves
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active May 26th 2010

polished and expanded (infinity,1)-category of (infinity,1)-sheaves

In particular I spelled out the proof that the full subcategory of (oo,1)-presheaves on (infinity,1)-sheaves is a left exact reflective sub-(oo,1)-category.

- Discussion Type
- discussion topicoo-Lie groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 25th 2010

started at infinity-Lie groupoid a section The (oo,1)-topos on CartSp.

Currently this gives statement and proof of the assertion that for a smooth manifold regarded as an object of $sPSh(CartSp)_{proj,cov}$ the Cech nerve of a

*good*open cover provides a cofibrant replacement.

- Discussion Type
- discussion topictopological localization at coverage
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 25th 2010

The discussion of topological localization and that at (infinity,1)-category of (infinity,1)-sheaves for obtaining (oo,1)-sheaf toposes focuses on Grothendieck topologies. In the rest of the nLab, though, we exhibit a certain moral preference for coverages.

I therefore started a section Localization at a coverage at model structure on simplicial presheaves, where I state and prove a handful of statements that are useful for understanding this.

There is more to be said here, but that’s it from me for the moment.

- Discussion Type
- discussion topicPoincare sphere
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active May 25th 2010

Wrote about Poincare sphere, which led to perfect group. Also added a subsection “Metrizable spaces” to metric space.