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 book bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie lie-theory 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 nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory 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 topicinfinitesimal interval object
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Oct 13th 2009

I have typed into infinitesimal interval object a detailed description of the simplicial object inuced on a microlinear space from the infinitesimal interval in immediate analogy to the construction of the finite path simplicial object induced from an interval object (as discussed there).

I also give the inclusion of the infinitesimal simplicial object into the finite one.

All the proofs here are straightforward checking, which I think I have done rather carefully on paper, but not typed up. What I would appreciate, though, is if somebody gave me a sanity check on the definition of the infinitesimal simplicial object (which is typed in detail).

In the very last section, which is the one that is still just a sketch, I am hoping to describe an isomorphism from my simplicial infinitesimal object to that considered by Anders Kock, which is currently described at infinitesimal singular simplicial complex in the case that the space X satisfies Kock's assumptions (it must be a "formal manifold").

The construction I discuss at infinitesimal interval object is supposed to generalize Kock's construction to all microlinear spaces and motivated by having that canonical obvious inclusion into the finite version at interval object.

The isomorphism should be evident: my construction evidently yields in degree k k-tuples of pairwise first oder neighbours if the space X admits that notion. But I want to sleep over this statement one more night...

- Discussion Type
- discussion topicQuestions on foundations
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 13th 2009

Roger Witte asks a question at foundations that looks interesting but which I haven't really thought about yet.

- Discussion Type
- discussion topicTodd wrote …
- Category Latest Changes
- Started by TobyBartels
- Comments 8
- Last comment by Todd_Trimble
- Last Active Oct 12th 2009

- Discussion Type
- discussion topicstructural meaning of [[axiom of foundation]]
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 12th 2009

I added a paragraph on structural meaning to axiom of foundation.

- Discussion Type
- discussion topicnLab on nLab
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Oct 12th 2009

I added the Lab itself to Online Resources, since that list is getting a lot of attention and may well be copied to other contexts.

- Discussion Type
- discussion topicRedirection redirection
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 11th 2009

I moved the redirect for de Rham cohomology from differential form to de Rham complex.

- Discussion Type
- discussion topic[[Gram-Schmidt process]]
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 10th 2009

Todd just wrote Gram-Schmidt process; I added a bit.

- Discussion Type
- discussion topicMoved biography to people
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Urs
- Last Active Oct 10th 2009

As planned here

- Discussion Type
- discussion topicNew pages
- Category Latest Changes
- Started by TobyBartels
- Comments 11
- Last comment by Mike Shulman
- Last Active Oct 10th 2009

pairing — pretty simple, but not to be confused with the product

- Discussion Type
- discussion topicTo go with [[simplicial category]] …
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 10th 2009

… we now have globular category.

- Discussion Type
- discussion topicsuper smooth topos
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by TobyBartels
- Last Active Oct 10th 2009

created super smooth topos

- Discussion Type
- discussion topiccircle of entries surrounding infinitesimal neighour
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Eric
- Last Active Oct 9th 2009

started infinitesimal neighbour and began creating a circle of entries surrounding this:

infinitesimal path infinity-groupoid in a smooth topos; path infinity-groupoid in a smooth topos; simplex in a lined topos

This is heading in the direction of giving a full discussion of for X a microlinear space, mentioned presently already at differential forms in synthetic differential geometry. I though i could just point to the literature for that, but not quite, apparently. Anders Kock discusses this for X a "formal manifold", an object with a cover by Kock-Lawvere vector spaces. But it should work a bit more generally using microlinear spaces, as indicated in the appendix of Models for Smooth Infinitesimal Analysis. There is an obvious general-nonsense definition wich I discuss, but I need yet to insert discussion of that and how this reproduces Kock's definition (but I think it does).

It has been an esteemed insight for me that the best way to think of all these constructions of "combinatorial differential forms" (still have to expand the discussion of those at differential forms in synthetic differential geometry) is by organizing them into their natural simplicial structures and then noticing that the model category structure available in the background allows us to think of the resulting simplicial objects in the topos as interna oo-groupoids. I think this must clearly the nLab way of thinking about this, so I created entries with the respective titles.

You may have noticed that on my personal web I am developing the further step that goes from (infinitesimal) path oo-groupoids of objects in a 1-topos to (infinitesimal) path oo-groupoids of objects in a "smooth (oo,1)-topos". I don't want to impose that fully (oo,1)-material on the main nLab as yet, before this is further developed, but the bits now added that just have oo-groupoids of paths in a 1-topos object is straightforward enough to warrent discussion here. i think.

While working on this, I also did various minor edits on the synthetic differential geometry context cluster, such as

splitting off lined topos from smooth topos

rewriting the introduction at Models for Smooth Infinitesimal Analysis (the previous remarks are by now better explained in the corresponding sub-entries and the new summary is supposed to get the main message of the book across better). Also created section headers there for each of the single models, which I hope I'll eventually describe there in a bit more detail each. Those toposes and they have there are mighty cool, I think, giving not only a well-adapted model for SDG but on top of that an implementation of nonstandard analysis, and of distribution theory. I am thinking that the toposophers among my co-laborants might enjoy looking at their smooth natural number object in a bit more. It's fun and seems to be much more relevant than seems to be widely appreciated.

Notice that at simplex in a lined topos I am asking for a reference. It's this standard construction of simplices as collpsed cylinders on lower dim simplicies. I don't think I should re-invent that wheel. What's the canonical reference for this general construction?

Finally please notice that all entries mentioned above are more or less stubby for the moment and need more work. But I thought it was about time to drop a latest-changes alert here now, before waiting longer.

- Discussion Type
- discussion topicsemiotic information
- Category Latest Changes
- Started by JonAwbrey
- Comments 1
- Last comment by JonAwbrey
- Last Active Oct 9th 2009

Inspired by David Corfield's blog entries on information geometry, I added a 'blink on semiotic information that I hope to develop over time.

- Discussion Type
- discussion topicpolsihing: infinitesimal object
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Oct 9th 2009

I tried to prettify the entry infinitesimal object:

I expanded and restructured the "Idea" section. I tried to emphasize the point that Lawvere's axioms are the right general point of view and that the wealth of constructions in algebraic geoemtry is, from this abstract nonsense point of view, to be regarded as taking place in a

*model*for these axioms. I cite Anders Kocks's latest book for the most simple minded version of how algebraic geometry is a model for sdg, but I think this goes through for more sophisticated versions, too. It would be nice to discuss this eventually elsewhere in some entry on "algebraic geometry as models for smooth toposes".I have also tried to subsume the approach of nonsstandard analysis as yet another special case of Lawvere's general axioms, by referring to Moerdijk-Reyes' topos and in which "objects of invertible infinitesimals/infinities" exist and model nonstandard analysis. This, too, would deserve being expanded on further, but I am thinking for the nLab this abstract-nonsense-first perspective is the right one.

Then I inserted some links to the now separate infinitesiaml interval object that I am still working on. I also changed the ideosyncratic terminology "infinitesimal k-cube" and "infinitsimal k-disk" to "cartesian product of inf. intervals" and "k-dimensional infinitesimal interval". Anders Kock calls the latter a "monad", following Leibniz, but I am hesitating to overload monad this way, given that Kock's use of it doesn't seem to be wide spread.

- Discussion Type
- discussion topicinfinitesimal singular simplicial object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 9th 2009

Spent all day with being distracted from this single thing that I planned to finish this morning: now at least a rough sketch is done

at infinitesimal interval object in the last section with the long section name I mean to define the "infinitesimal singular simplicial complex" in a new way.

Anders Kock defines this guy for "formal manifols", roughly, for spaces that have an atlas by vector spaces. There the simple definition applies recalled at infinitesimal singular simplicial complex.

But there should be a definition for arbitrary microlinear spaces, And it should be such that it is almost manfestly the inifnitesimal version of the path oo-groupoid construction described at interval object. This is what I am aiming to describe here.

One crucial thing is that we want that morphisms out of the objects in degree k of the infinitesimal singular simplicial complex that vanish on degenerate k-simplices are automatically fiberwise skew-linear. Seeing this in the construction that I am presenting there seems to be different to the way Anders Kock describes it in the other setup. This is the main thing I need to check again when i am more awake.

- Discussion Type
- discussion topichomotopy - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 8th 2009

I created homotopy - contents and added it as a floating table of contents to relevant entries.

This was motivated from the creation of infinitesimal interval object and the desire to provide a kind of map that shows how that concept sits in the greater scheme of things. The homotopy - contents was supposed to be a step in that diretion.

I really meant to expand at infinitesimal interval object on something I already meant to provide yesterday, but then that table of contents kept distracting me, and the fact that I came across mapping cone while editing it and felt compelled to improved that entry first, which I did

- Discussion Type
- discussion topicinterval object: induced path oo-groupoids
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Oct 8th 2009

I removed my recent material at simplex in a lined topos and instead inserted this now, expanded, at

where it belongs. There is now a section there that discusses how interval objects gives rise to cubical and simplicial path oo-categories.

The proposition I state there I have carefully checked. Should be correct. But haven't typed the proof, it doesn't lend itself to being typed (straightforward but tedious, as one says).

But if it is indeed correct, this must be standard well-known stuff. Does anyone have a reference?!

I also restructured and edited the rest of the entry a bit, trying to make it a bit nicer. THis entry deserves more attention, it is a pivotal entry.

Tomorrw when I am more awake I'll remove simplex in a lined topos and redirect links to it suitably to interval oject.

- Discussion Type
- discussion topichowto and faq
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 8th 2009

I moved the instructions on making diagrams from FAQ to HowTo, which seemed a better fit, and added a comment about including images as another method. I also made the individual questions at FAQ into ### headers, rather than numbered lists, so that they would show up in the automatic table of contents.

- Discussion Type
- discussion topicsmooth loci
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Oct 8th 2009

created stub for smooth loci

(should it be "smooth locus" instead?)

- Discussion Type
- discussion topicgeneralized smooth algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 7th 2009

sty addition to generalized smooth algebra: remark on terminology added and section on "internal definition" added.

planning to polish thinmgs later

- Discussion Type
- discussion topicIsrael Gelfand
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 7th 2009

created entry on Israel Gelfand with the material that John posted to the blog.

turns out the "Timeline" entry was already requesting it

- Discussion Type
- discussion topicmicrolinear space
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by TobyBartels
- Last Active Oct 6th 2009

created microlinear space

One thing I might be mixed up above:

in the literature I have seen it seems to say that

$ X^D x_X X^D \simeq X^{D(2)}$

with

$ D(2) = { (x_1,x_2) \in R \times R | x_i x_j = 0} $.

But shouldn't it be

$ D(2)' = { (x_1,x_2) \in R \times R | x_i^2 = 0} $.

?

- Discussion Type
- discussion topicsynthetic differential geometry - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 6th 2009

started synthetic differential geometry - contents and added it as floating table of contents to the relevant entries

- Discussion Type
- discussion topicinfinitesimal space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 6th 2009

quick addition of "formal infinitesimal spaces" and Weil algebras to infinitesimal space

but am planning to polish this entry further later, it is a bit of a mess at the moment

- Discussion Type
- discussion topicintegration axiom
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 6th 2009

started integration axiom, but incomplete for the moment

at schreiber:integration of oo-Lie algebroid valued differential forms I am thinking about the higher version of this

- Discussion Type
- discussion topiccyclic things
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by David_Corfield
- Last Active Oct 5th 2009

Looks like maybe Todd is right at cyclic order that the cycle category cannot be defined in that way.

- Discussion Type
- discussion topicAdded detexify to online resources
- Category Latest Changes
- Started by Andrew Stacey
- Comments 1
- Last comment by Andrew Stacey
- Last Active Oct 5th 2009

Just learnt about detexify from the Secret Blogging Seminar and thought it worth adding to the online resources page.

- Discussion Type
- discussion topicNew pages
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Oct 5th 2009

- Discussion Type
- discussion topicnatural isomorphism
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Oct 4th 2009

A little more detail at natural isomorphism, including when one can speak of the functor satisfying certain conditions.

- Discussion Type
- discussion topicsynthetic differential geometry
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 4th 2009

I worked on synthetic differential geometry:

I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.

Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.