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-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar 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 stack string string-theory 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 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.

- 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.