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- 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 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.

- 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 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 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 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.

- Discussion Type
- discussion topicspecial relativity
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Ian_Durham
- Last Active May 19th 2010

somebody signing as “Anonymous Coward” had created special relativity and typed in a confused paragraph (the smallest confusion being that the paragraph concerned not special but general relativity).

I removed that paragraph and quickly wrote a brief “Idea”-section . But have no time to do this justice now.

- Discussion Type
- discussion topic(oo,1)-site
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 19th 2010

created (infinity,1)-site

- Discussion Type
- discussion topicfinite limit
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active May 19th 2010

created finite limit (this was previously a redirect to finitely complete category)

- Discussion Type
- discussion topicSimple groups
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active May 19th 2010

It got announced in another category, but here it is in Latest Changes:

Todd began (and then I edited) simple group.

- Discussion Type
- discussion topicconformal group
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 19th 2010

I wrote a quick entry conformal group, just from memory. Somebody could check and expand. In fact it would not be bad to have also a separate entry on conformal and on quasiconformal mappings.

- Discussion Type
- discussion topicchiral algebra
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 19th 2010

chiral algebra and improvements to vertex operator algebra

- Discussion Type
- discussion topicduplicate page compact operators
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 8
- Last comment by zskoda
- Last Active May 18th 2010

somehow I missed that there already is a page compact operator and created compact operators. The plural is another error :-) the unsatisfied link that I used to create the page was “compact operators”. When I tried to rename it to the singular term it failed, of course. Now the page compact operators is simply superfluous, but as a non-administrator I cannot delete it…

- Discussion Type
- discussion topicsequential compactness
- Category Latest Changes
- Started by Andrew Stacey
- Comments 31
- Last comment by Andrew Stacey
- Last Active May 18th 2010

Created sequential compactness, should probably link to all these compactness variations from compact space. Not sure if I got the “iff” bit right in the relationship with compactness itself; will check it myself if no-one fixes it in the meantime.

I decided that this was the key property in manifolds of mapping spaces and to stop trying to figure out a Froelicher version of sequentially compact for the time-being.

- Discussion Type
- discussion topictwist
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active May 17th 2010

the term “twist” or “twisted” is one of those insanely ambiguous terms in math. Trying to follow our recent agreement on how to deal with ambiguous page names, I tried to indicate this at twist .

- Discussion Type
- discussion topicLax-idempotent monads
- Category Latest Changes
- Started by FinnLawler
- Comments 10
- Last comment by Mike Shulman
- Last Active May 17th 2010

Created lax-idempotent 2-monad, with some definitions from Kelly–Lack. I think Kock has a couple of others. I’ll add more, like proofs of the equivalence of the definitions, and more on the cocompletion example, later (next week, probably).

- Discussion Type
- discussion topicFinal version of thesis
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by Harry Gindi
- Last Active May 17th 2010

The final copy of my thesis is up on the lab. Available from Fundamental Bigroupoids and 2-Covering Spaces. I’ve fixed the typo in definition 5.1 that made it into the print copy ;)

Thus I’ve updated the links at David Roberts, the above linked page, and on my private web home page. If anyone knows of any other places it is linked, please let me know, or update the link to point to DMRthesis_final.pdf, instead of DMR_thesis.pdf.

Now to all the other projects that are on the back burner, time permitting…

- Discussion Type
- discussion topicIntroductions to category theory in physics
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Ian_Durham
- Last Active May 17th 2010

started a section Introductions to category theory in physics at the woefully imperfect entry higher category theory and physics. So far this contains mostly th expository articles by Bob Coecke.

- Discussion Type
- discussion topicpresheaves and overcategories
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by DavidRoberts
- Last Active May 16th 2010

I put the theorem about presheaves on overcategories and overcategories of presheaves that had its own page at functors and comma categories into the Properties-section at category of presheaves: Presheaves on over-categories and over-categories of presheaves.

Then I added the analogous proposition for (oo,1)-presheaves at (∞,1)-category of (∞,1)-presheaves -- Interaction with overcategories

Incidentally, there is some bug on the nLab that might be related to the one Toby just pointed out in the thread on scrollboxes: Trying to put links to subsections of nLab entries into nLab entries is often troublesome. The Markup-code for links gets mixed up by the hash-sign, usually. Then usually the html-code will work. But at the moment at category of presheaves I cant get that to work either...

- Discussion Type
- discussion topicmonoidal categories sidebar
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Urs
- Last Active May 16th 2010

Prompted by Peter Selinger’s recent email on the catlist, I created a floating TOC for monoidal categories, added it to a lot of pages, created a couple of stubs for ribbon category and pivotal category, and corrected the redirect for autonomous category to point to rigid monoidal category rather than compact closed category. We are still missing stubs for balanced monoidal category and traced monoidal category and dagger monoidal category – anyone want to fill them in?

- Discussion Type
- discussion topictoc for topos theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 14th 2010

started a floating toc for topos theory. See at the right of topos.

Please feel encouraged to expand and improve the structure.