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- 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 topicsyntomic cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 28th 2010

stub for syntomic cohomology

- 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 topicpath space object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 26th 2010

added to path space object an Examples-section with some model category-theoretic discussion, leading up to the statement that in a simplicial model category for fibrant $X$ the powering $X^{\Delta[1]}$ is always a path space object.

- 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.

- Discussion Type
- discussion topicOnline resource
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active May 24th 2010

Added Manifold Atlas Project to Online Resources.

- Discussion Type
- discussion topictriangulable spaces
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active May 24th 2010

I added a section on triangulable spaces and PL structures to simplicial complex, but this is the type of thing which gets beyond my ken pretty quickly. My real motivation is to convince myself that a space is homeomorphic to the realization of a simplicial complex (in short, is triangulable) if and only if it is homeomorphic to the realization of a simplicial set – perhaps this seems intuitively obvious, but it should be given a careful proof, and I want such a proof to have a home in the Lab. (Tim Porter said in a related discussion that there was a relevant article by Curtis in some early issue of Adv. Math., but I am not near a university library to investigate this.)

I’ll put down some preliminary discussion here. Let $P_{fin}(X)$ denote the poset of finite

*nonempty*subsets of $X$. A simplicial complex consists of a set $V$ and a down-closed subset $\Sigma \subseteq P_{fin}(V)$ such that every singleton $\{v\}$ belongs to $\Sigma$. Thus $\Sigma$ is itself a poset, and we can take its nerve as a simplicial set. The first claim is that the realization of this nerve is homeomorphic to the realization of the simplicial complex. This I believe is or should be a basic result in the technique of subdivision. Hence realizations of simplicial sets subsume triangulable spaces.For the other (harder) direction, showing that realizations of simplicial sets are triangulable, I want a lemma: that the realization of a nerve of a poset is triangulable. Basically the idea is that we use the simplicial complex whose vertices are elements of the poset and whose simplices are subsets $\{x_1, x_2, \ldots, x_n\}$ for which we have a strictly increasing chain $x_1 \lt x_2 \lt \ldots \lt x_n$. Then, the next step would use the following construction: given a simplicial set $X$, construct the poset whose elements are

*nondegenerate*simplices (elements) of $X$, ordered $x \lt y$ if $x$ is some face of $y$. The claim would be that the realization of $X$ is homeomorphic to the realization of the nerve of this poset.All of this could very well be completely standard, but it’s hard for me to find an account of this in one place. Alternatively, my intuitions might be wrong here. Or, perhaps I’m going about it in a clumsy way.

- Discussion Type
- discussion topicVistoli, Notes on Grothendieck topologies, fibered categories and descent theory
- Category Latest Changes
- Started by Eric
- Comments 32
- Last comment by Urs
- Last Active May 24th 2010

I started adding some illustrations to my personal web related to Vistoli’s paper on descent. If you like them or have suggestions to improve them, I can maybe migrate some to nLab pages:

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory (ericforgy)

- Discussion Type
- discussion topicCurrying
- Category Latest Changes
- Started by TobyBartels
- Comments 17
- Last comment by Mike Shulman
- Last Active May 24th 2010

Todd Trimble requested currying (on the Sandbox, of all places), and I wrote it (also linking to it from closed monoidal category).

- Discussion Type
- discussion topicOperad
- Category Latest Changes
- Started by Harry Gindi
- Comments 3
- Last comment by Harry Gindi
- Last Active May 23rd 2010

So, I have some pending changes on operad that I made in the sandbox and am waiting for a go-ahead to post from the interested parties, but I was also wondering if someone would be willing to write up a follow-up to the very nice definition of an operad as a monoid in the blah blah monoidal category. That is, it seems like this should give us a very nice way to define an algebra, but I don't know how one would actually go about doing it.

- Discussion Type
- discussion topiccategory theory - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2010

rearranged a bit and expanded category theory - contents. In particular I added a list with central theorems of category theory.

- Discussion Type
- discussion topicYoneda lemma - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2010

created sidebar toc Yoneda lemma - contents.

Not yet convinced myself that I found the right subdivisions and probably forgot some entries. Please improve.

- Discussion Type
- discussion topicrepresentable presheaf
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2010

added Eric’s illustrations to the Idea-section at representable presheaf. Also added a stub-section on Definition in higher category theory.

- Discussion Type
- discussion topickinds of morphisms in 2-categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Mike Shulman
- Last Active May 21st 2010

Created faithful morphism, conservative morphism, pseudomonic morphism, and discrete morphism, and added to fully faithful morphism.

- Discussion Type
- discussion topichom-functor
- Category Latest Changes
- Started by Eric
- Comments 11
- Last comment by Eric
- Last Active May 21st 2010

On the page hom-functor, it says

There is also a

$hom(-,c) : C^{op} \to Set,$**contravariant hom-functor**where $C^{op}$ is the opposite category to $C$, which sends any object $x \in C^{op}$ to the hom-set $hom(x,c)$.

If you write it like this, should you really call it “contravariant”? When you write $C^{op}$, I thought you should call it just “functor” or “covariant”. By saying it is contravariant AND writing $C^{op}$, it seems like double counting.

I hope to add some illustrations to these pages. It is a shame there are not more illustrations on the nLab since nStuff is so amenable to nice pictures.

- Discussion Type
- discussion topicMore fun with functional analysis: complete and normable
- Category Latest Changes
- Started by Andrew Stacey
- Comments 21
- Last comment by Andrew Stacey
- Last Active May 21st 2010

Added complete topological vector space including various variants (quasi-complete, sequentially complete, and some others). Hopefully got all the redirects right!

I only have Schaefer’s book at home so couldn’t check “locally complete” - I know that Jarchow deals with this in his book. Kriegl and Michor naturally only consider it in the context of smootheology so I’m not sure what the “best” characterisation is. There’s also a notational conflict with “convenient” versus “locally complete”. As Greg Kuperberg pointed out, in some places “convenient” means “locally complete and bornological” whereas in others it means just “locally complete” (in the contexts where convenient is used the distinction is immaterial as the topology is not considered an integral part of the structure).

I added these whilst working on the expansion of the TVS relationships diagram. That brought up a question on terminology. In the diagram, we have entries “Banach space” and “Hilbert space” (and “normed space” and “inner product space”). These don’t quite work, though, as for a topological vector space the correct notion of a normed space should be

*normable*space as the actual choice of norm is immaterial for the TVS properties. I’m wondering whether or not this is something to worry about. Here’s an example of where it may be an issue: a nuclear Banach space is automatically finite dimensional. That implies that its topology can be given by a Hilbert structure. However, the Hilbertian norm may not be the one that was first thought of. But that’s a subtlety that’s tricky to convey on a simple diagram. So I’d rather have “normable” than “normed”. Does anyone else have an opinion on this?If “normable” is fine, then the important question is: what’s a better way of saying “Hilbertisable”, or “Banachable”? Length doesn’t matter here, as I’m putting the expanded names in tooltips and only using abbreviations in the diagram.

- Discussion Type
- discussion topiccodiscrete cofibration
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Mike Shulman
- Last Active May 21st 2010

I started writing something about codiscrete cofibrations, which is a nice way that many categories can be canonically equipped with proarrows. Richard Garner is visiting Chicago this week, and yesterday some of us were talking about how this construction can be made very functorial, giving a very nice way to construct functors and monads on equipments; I plan to add this to the entry as well.

- Discussion Type
- discussion topicrational homotopy theory in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active May 20th 2010

started rational homotopy theory in an (infinity,1)-topos

With just slightly more it could also be called "Lie theory in an oo,1-topos" I suppose.

if you looked at this yesterday, as it was under construction, maybe have another look: I believe I could clarify the global story a bit better.

- Discussion Type
- discussion topicQuery about finite dimensional Banach spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 12
- Last comment by Todd_Trimble
- Last Active May 20th 2010

Looking at the entry Banach spaces, I found the following in the introduction:

So every $n$-dimensional real Banach space may be described (up to linear isometry, the usual sort of isomorphism) as the Cartesian space $\mathbb{R}^n$ equipped with the $p$-norm for $1 \leq p \leq \infty$

which seems to imply that every norm on a finite dimensional Banach space is a $p$-norm for some $p$. That feels to me like a load of dingo’s kidneys. To define a norm on some $\mathbb{R}^n$ I just need a nice convex set, and there’s lots more of these than the balls of $p$-norms, surely.

Am I missing something?

- Discussion Type
- discussion topicMoonshine
- Category Latest Changes
- Started by zskoda
- Comments 25
- Last comment by zskoda
- Last Active May 20th 2010

Moonshine, intentionally with capital M as most people do follow this convention for the Monster and (Monstrous) Moonshine VOA.