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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion 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 digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad 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 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 tqft type type-theory universal variational-calculus

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- Discussion Type
- discussion topiccritical loci in differential cohesive homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 2nd 2012

I have started making notes at

*differential cohesion*on the axiomatic formulation ofSo far just the bare basics. To be expanded…

The basic observation (easy in itself, but fundamental for the concept formation) is that for any differential cohesive homotopy type $X$, the inclusion of the formally étale maps into $X$ into the full slice over $X$ is not only reflective but also co-reflective (since the formally étale maps are the Pi_inf-closed morphisms with the infinitesimal path groupoid functor / de Rham space functor $\Pi_{inf}$ being a left adjoint).

This means that for $G$ any differential cohesive $\infty$-group with the corresponding de Rham coefficient object $\flat_{dR}\mathbf{B}G$ (the universal moduli for flat $\mathfrak{g}$-valued differential forms), the sheaf of flat $\mathfrak{g}$-valued forms over any $X$ is given by the sections of the coreflection of the product projection $X \times \flat_{dR}\mathbf{B}G \to X$ into the formally étale morphisms into $X$.

- Discussion Type
- discussion topicrealizability model
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 2nd 2012

I created an entry

*realizability model*. But I only got to put one single reference into it and now I am forced to go offline.I’ll try to add more later. But maybe somebody here feels inspired to add a brief explanation…

- Discussion Type
- discussion topicHOMFLY
- Category Latest Changes
- Started by Andrew Stacey
- Comments 5
- Last comment by Tim_Porter
- Last Active Dec 1st 2012

Added HOMFLY-PT polynomial. Hope I got the skein diagrams right!

- Discussion Type
- discussion topic(oo,1)-quasitopos
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Mike Shulman
- Last Active Nov 30th 2012

started (infinity,1)-quasitopos

- Discussion Type
- discussion topiccomputational consistency
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 30th 2012

not sure why, but reading

- Peter Dybjer,
*Thoughts on Martin-Löf’s Meaning Explanations*(pdf)

made me look at

- Alexandre Miquel,
*The experimental effectiveness of mathematical proof*(pdf)

which seems to be about something deep and important that eventually I’d like to better grasp (but don’t yet), and that made me create

*computational consistency*.But I admit that I don’t really know what I am doing, in this case. So I’ll stop.

- Peter Dybjer,

- Discussion Type
- discussion topicdescent along torsors
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 29th 2012

New entries descent along a torsor and Schneider’s descent theorem. Some changes and literature additions to a number of related entries.

- Discussion Type
- discussion topicaction functional
- Category Latest Changes
- Started by TobyBartels
- Comments 22
- Last comment by Urs
- Last Active Nov 28th 2012

Added some formulas and a manifestly relativistic version to action functional.

I have also been reverting JA's changes to variant conventions of spelling and grammar.

- Discussion Type
- discussion topic(infinity,n)-topos / (infinity,n)-sheaf
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Nov 28th 2012

started stubs for

*(infinity,n)-sheaf*and*(infinity,n)-topos*; for the moment mostly as receptors and donors of cross-links, only.

- Discussion Type
- discussion topicinternal (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 60
- Last comment by DavidRoberts
- Last Active Nov 28th 2012

I am starting an entry internal (infinity,1)-category about complete Segal-like things.

This is prompted by me needing a place to state and prove the following assertion: a cohesive $\infty$-topos is an “absolute distributor” in the sense of Lurie, hence a suitable context for internalizing $(\infty,1)$-categories.

But first I want a better infrastructure. In the course of this I also created a “floating table of contents”

and added it to the relevant entries.

- Discussion Type
- discussion topiccore in a 2-category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Mike Shulman
- Last Active Nov 28th 2012

I have taken the liberty of moving one more bit of Mike’s stuff on 2-topos theory from his personal web to the main $n$Lab, namely

*core in a 2-category*.(I am trying to fill what used to be the gray links in the proof at

*2-topos – In terms of internal categories*).

- Discussion Type
- discussion topicn-localic 2-topos
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Mike Shulman
- Last Active Nov 28th 2012

I have taken the liberty of moving one more bit of Mike’s stuff on 2-topos theory from his personal web to the main $n$Lab, namely

*n-localic 2-topos*.(I am trying to fill what used to be the gray links in the proof at

*2-topos – In terms of intenral categories*).

- Discussion Type
- discussion topicBn-geometry
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Todd_Trimble
- Last Active Nov 27th 2012

note on

*Bn-geometry*

- Discussion Type
- discussion topicweak omega-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 27th 2012

Put a link to

- Thorsten Altenkirch, Ondrej Rypácek,
*A Syntactical Approach to Weak $\omega$-Groupoids*(pdf)

into

*weak omega-groupoid*… only trouble being that this entry doesn’t exist yet but redirects to*infinity-groupoid*, which otherwise has no references currently ?!-o . Somebody should take care of editing this a bit. But it won’t be me right now.- Thorsten Altenkirch, Ondrej Rypácek,

- Discussion Type
- discussion topicroot
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Nov 26th 2012

I made some modifications to the definition section of root, and added the theorem that finite multiplicative subgroups of a field are cyclic. While I was at it, I added a bit to quaternion.

- Discussion Type
- discussion topicZuckerman induction
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 26th 2012

I have split off a brief entry

*Zuckerman induction*from*cohomological induction*(since the basic version is not necessarily derived).

- Discussion Type
- discussion topicA-infinity space
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Nov 25th 2012

In email discussion with somebody I wanted to point to the $n$Lab entry

*A-infinity space*only to notice that there is not much there. I have now spent a minute adding just a tiny little bit more…

- Discussion Type
- discussion topiccohomological induction
- Category Latest Changes
- Started by zskoda
- Comments 14
- Last comment by Urs
- Last Active Nov 25th 2012

Idea section for a new entry cohomological induction and a new stub induced comodule. I have separated corepresentation from comodule&coaction. Sometimes corepresentation is the same as coaction, sometimes there are small differences (defined on dense subspaces etc.) but more important, there is a different notion of corepresentation in Leibniz algebra theory, which will be explained in a separate section later.

A remark at induction.

- Discussion Type
- discussion topicmodel structure for homotopy n-types
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 24th 2012

I could have sworn that we already had the following entry, but it seems we didn’t. Now we do:

- Discussion Type
- discussion topicinfinity-image
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active Nov 22nd 2012

As already mentioned in another thread, I have added to

*infinity-image*a brief new section*Syntax in homotopy type theory*. But please check! And even if correct, it’s still a bit rough.

- Discussion Type
- discussion topichigher order sharp-concrete objects
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2012

It’s time to become serious about “higher order” aspects of applications of the the “sharp-modality” $\sharp$ in a cohesive (infinity,1)-topos $\mathbf{H}$ – I am thinking of the construction of moduli $\infty$-stacks for differential cocycles.

Consider, as usual, the running example $\mathbf{H} = Sh_\infty(CartSp) =$ Smooth∞Grpd.

## Simple motivating example: moduli of differential forms

Here is the baby example, which below I discuss how to refine:

there is an object called $\Omega^1 \in \mathbf{H}$, which is just the good old sheaf of differential 1-forms. Consider also a smooth manifold $X \in \mathbf{H}$. On first thought one might want to say that the internal hom object $[X, \Omega^1]$ is the “moduli 0-stack of differential 1-forms on $X$”. But that’s not quite right. For $U \in$ CartSp, the $U$-plots of the latter should be smoothly $U$-parameterized sets of differential 1-forms on $X$, but the $U$-plots of $[X,\Omega^1]$ contain a bit more stuff. They are of course 1-forms on $U \times X$ and the actual families that we want to see are only those 1-forms on $U \times X$ which have “no leg along $U$”. But one sees easily that the correct moduli stack of 1-forms on $X$ is

$\mathbf{\Omega}^1(X) := \sharp_1 [X,\Omega^1] \hookrightarrow \sharp [X, \Omega^1] \,,$where $\sharp_1 [X,\Omega^1] := image( [X, \Omega^1] \to \sharp [X, \Omega^1] )$ is the

*concretification*of $[X,\Omega^1]$.## Next easy example: moduli of connections

This above kind of issue persists as we refine differential 1-forms to circle-principal connections: write $\mathbf{B}U(1)_{conn} \in \mathbf{H}$ for the stack of circle-principal connections. Then for $X$ a manifold, one might be inclined to say that the mapping stack $[X, \mathbf{B}U(1)_{conn}]$ is the moduli stack of circle-principal connections on $X$. But again it is not quite right: a $U$-plot of $[X,\mathbf{B}U(1)_{conn}]$ is a circle-principal connection on $U \times X$, but it should be one with no form components along $U$, so that we can interpret it as a smoothly $U$-parameterized set of connections on $X$.

The previous example might make one think that this is again fixed by considering $\sharp_1 [X, \mathbf{B}U(1)_{conn}]$. But now that we have a genuine 1-stack and not a 0-stack anymore, this is not good enough: the stack $\sharp_1 [X, \mathbf{B}U(1)_{conn}]$ has as $U$-plots the groupoid whose objects are smoothly $U$-parameterized sets of connections on $X$ – that’s as it should be – , but whose morphisms are $\Gamma(U)$-parameterized sets of gauge transformations between these, where $\Gamma(U)$ is the underlying discrete set of the test manifold $U$ – and that’s of course not how it should be. The reflection $\sharp_1$ fixes the moduli in degree 0 correctly, but it “dustifies” their automorphisms in degree 1.

We can correct this as follows: the correct moduli stack $U(1)\mathbf{Conn}(X)$ of circle principal connections on some $X$ is the homotopy pullback in

$\array{ U(1)\mathbf{Conn}(X) &\to& [X, \mathbf{B} U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B} U(1)] }$where the bottom morphism is induced from the canonical map $\mathbf{B}U(1)_{conn} \to \mathbf{B}U(1)$ from circle-principal connections to their underlying circle-principal bundles.

Here the $\sharp_1$ in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified $\Gamma(U)$-parameterized sets of gauge transformations to those that actually do have a smooth parameterization.

## More serious example: moduli of 2-connections

The previous example is controlled by a hidden pattern, which we can bring out by noticing that

$[X, \mathbf{B}U(1)] \simeq \sharp_2 [X, \mathbf{B}U(1)]$where $\sharp_2$ is the 2-image of $id \to \sharp$, hence the factorization by a 0-connected morphism followed by a 0-truncated one. For the 1-truncated object $[X, \mathbf{B}U(1)]$ the 2-image doesn’t change anything. Generally we have a tower

$id = \sharp_\infty \to \cdots \to \sharp_2 \to \sharp_1 \to \sharp_0 = \sharp \,.$Moreover, if we write $DK$ for the Dold-Kan map from sheaves of chain complexes to sheaves of groupoids (and let stackification be implicit), then

$\begin{aligned} \mathbf{B}U(1)_{conn} &= DK( U(1) \to \Omega^1 ) \\ \mathbf{B}U(1) &= DK( U(1) \to 0 ) \end{aligned} \,.$If we pass to circle-principal 2-connections, this becomes

$\begin{aligned} \mathbf{B}^2 U(1)_{conn^1} = \mathbf{B}^2U(1)_{conn} &= DK( U(1) \to \Omega^1 \to \Omega^2 ) \\ \mathbf{B}U(1)_{conn^2} & = DK( U(1) \to \Omega^1 \to 0 ) \\ \mathbf{B}U(1)_{conn^3} = \mathbf{B}^2 U(1) & = DK( U(1) \to 0 \to 0 ) \end{aligned}$and so on.

And a little reflection show that the correct moduli 2-stack $(\mathbf{B}U(1))\mathbf{Conn}(X)$ of circle-principal 2-connections on some $X$ is the homotopy limit in

$\array{ (\mathbf{B}U(1))\mathbf{Conn}(X) &\to& &\to& [X, \mathbf{B}^2 U(1)] \\ && && \downarrow \\ && \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^2}] &\to& \sharp_2 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B}^2 U(1)_{conn^2}] } \,.$This is a “3-stage $\sharp$-reflection” of sorts, which fixes the naive moduli 2-stack $[X, \mathbf{B}^2 U(1)]$ first in degree 0 (thereby first completely messing it up in the higher degrees), then fixes it in degree 1, then in degree 2. Then we are done.

- Discussion Type
- discussion topicBondal-Orlov reconstruction theorem
- Category Latest Changes
- Started by adeelkh
- Comments 3
- Last comment by adeelkh
- Last Active Nov 20th 2012

I just added a page on the Bondal-Orlov reconstruction theorem. Feel free to edit!

- Discussion Type
- discussion topicFraenkel model
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by DavidRoberts
- Last Active Nov 19th 2012

It should have its own announcement: Frankel model of ZFA added to the lab. I should say that in this model there is a map $x \to \mathbb{N}$ where every fibre has two elements, which has no section (which would be “choosing a sock out of a countable set of pairs”)

- Discussion Type
- discussion topicAbraham Fraenkel
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Nov 16th 2012

Created Abraham Fraenkel, the F in ZFC.

- Discussion Type
- discussion topicHamiltonian vector field
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Nov 13th 2012

I have splitt off Hamiltonian vector field from symplectic manifold in order to also record the $n$-plectic generalization.

- Discussion Type
- discussion topicFaà di Bruno formula
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 13th 2012

Faà di Bruno formula with redirect Faà di Bruno Hopf algebra

- Discussion Type
- discussion topicinfinitesimal Galois theory and differential modality
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by zskoda
- Last Active Nov 13th 2012

We had discussed here at some length the formalization of formally etale morphisms in a differential cohesive (infinity,1)-topos. But there is an immediate slight reformulation which I never made explicit before, but which is interesting to make explicit:

namely I used to characterize formal étaleness in terms of the canonical morphism $\phi : i_! \to i_*$ between the components of the geometric morphism $i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ that defines the differential cohesion – because that formulation made close contact to the way Kontsevich and Rosenberg formulate formal étaleness.

But there is a more suggestive/transparent but equivalent (in fact more general, since it works in all of $\mathbf{H}_{th}$ not just in $\mathbf{H}$) formulation in terms of the $\mathbf{\Pi}_{inf}$-modality, the “fundamental infinitesimal path $\infty$-groupoid” operator:

a morphism $f : X \to Y$ in $\mathbf{H}_{th}$ is formally étale precisely if the canonical diagram

$\array{ X &\to & \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(Y) }$is an $\infty$-pullback.

(It’s immediate that this is equivalent to the previous definition, using that $i_!$ is fully faithful, by definition.)

This is nice, because it makes the relation to general abstract Galois theory manifest: if we just replace in the above the infinitesimal modality $\mathbf{\Pi}_{inf}$ with the finite path $\infty$-groupoid modality $\mathbf{\Pi}$, then the above pullback characterizes the “$\mathbf{\Pi}$-closed morphisms” which precisely constitute the total space projections of locally constant $\infty$-stacks over $Y$. Here we now characterize

*general*$\infty$-stacks over $Y$.And for instance in direct analogy with the corresponding proof for the $\mathbf{\Pi}$-modality, one finds for the $\mathbf{\Pi}_{inf}$-modality that, for instance, we have an orthogonal factorization system

$(\mathbf{\Pi}_{inf}-equivalences\;,\; formally\;etale\;morphisms) \,.$I’ll spell out more on this at

*Differential cohesion – Structures*a little later (that’s why this here is under “latest changes”), for the moment more details are in section 3.7.3 of*differential cohomology in a cohesive topos (schreiber)*.

- Discussion Type
- discussion topicdisk
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 13th 2012

added to

*disk*a brief pointer to Joyal’s combinatorial disks. Needs to be expanded, probably entry should be split and disambiguated. But no time right now.

- Discussion Type
- discussion topiclayers of foundations
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by TobyBartels
- Last Active Nov 13th 2012

I have started something in an entry

which has grown out of the the desires expressed in the thread

*The Wiki history of the universe*.This is tentative. I should have maybe created this instead on my personal web. I hope we can discuss this a bit. If it leads nowhere and/or if the feeling is that it is awkward for one reason or other, I promise to remove it again. But let’s give this a chance. I feel this is finally beginning to converge to something.

- Discussion Type
- discussion topicnon-Hausdorff manifold
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by TobyBartels
- Last Active Nov 13th 2012

*non-Hausdorff manifold*(just for complenetess, since I was editing the exposition at*manifold*)

- Discussion Type
- discussion topicLawvere references
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by DavidRoberts
- Last Active Nov 13th 2012

started adding list of references to the page Bill Lawvere

not that I made it very far -- just three items so far :-)

I was really looking for an online copy of "Categorical dynamics" as referenced at synthetic differential geometry and generalized smooth algebra, but haven't found it yet. I was thinking that the "Toposes of laws of motion" that I do reference must be something close. But I don't know.

- Discussion Type
- discussion topicaffine algebra, affine variety
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 12th 2012

New, mainly disambiguation, entry affine algebra. Note that affine algebra for most algebraists is not the same as affine Lie algebra. I have corrected a wrong link in Wess-Zumino model which links to affine algebra instead to affine Lie algebra; let us be careful when linking in future. Affine algebras are coordinate rings of affine varieties. I have split affine variety from algebraic variety which also got a redirect algebraic manifold (= smooth algebraic variety). New entry Igor Shafarevich.

- Discussion Type
- discussion topicBargmann-Segal transform, Hall coherent state
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 12th 2012

New stubs Bargmann-Segal transform and Hall coherent state. Changes to coherent state and coherent state in geometric quantization. We still need Bergman kernel (the coincidence is that the (Segal-)Bargmann kernel is a special case of Bergman kernel from complex analysis :))

- Discussion Type
- discussion topicdifferential Galois theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 12th 2012

started

*differential Galois theory*

- Discussion Type
- discussion topic2-framing
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 12th 2012

expanded at

*2-framing*

- Discussion Type
- discussion topicAlex Heller
- Category Latest Changes
- Started by Tim_Porter
- Comments 18
- Last comment by Tim_Porter
- Last Active Nov 12th 2012

I created Alex Heller at Jim’s suggestion. It is very stubby and could have a lot more added.

- Discussion Type
- discussion topicAndree Ehresmann
- Category Latest Changes
- Started by DavidRoberts
- Comments 3
- Last comment by DavidRoberts
- Last Active Nov 12th 2012

New page for A. Ehresmann (and relevant redirects, including Bastiani) and links with Ehresmann, Cahiers.

- Discussion Type
- discussion topicZoran: new entries
- Category Latest Changes
- Started by zskoda
- Comments 250
- Last comment by zskoda
- Last Active Nov 12th 2012

- New entry discrete mathematics. Appropriate change at mathematicscontents. New entry multiset, Jordan algebra.

More will be added here as it goes, new items down.

- Discussion Type
- discussion topictangent Lie algebra, Chevalley group
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 11th 2012

New entry (!) tangent Lie algebra. Significant changes at invariant differential form with redirect invariant vector field reflecting the vector fields and other tensor cases. Many more related entries listed at and the whole entry reworked extensively at Lie theory. Some changes at Lie’s three theorems and local Lie group. New stubs Chevalley group and Sigurdur Helgason.

By the way, when writing tangent Lie algebra, I had the problem with finding the correct font for the standard symbol of Lie algebra of vector fields on a manifold. Usually one has varchi symbol which looks like Greek chi but with dash through middle. The varchi symbol is not recognized and I put mathcal X which is slanted and script, just alike, but without dash through middle.

By the way, on a real Lie group $G$ of dimension $n$, if one expresses the right invariant vector field in terms of left invariant vector fields then at each point there is a $\mathbb{R}$-linear operator which sends any frame of left invariant vector fields to the

*corresponding*frame in right invariant vector fields; this gives a $GL_n(\mathbb{R})$-valued real analytic function on $G$ (or, in local coordinates, on a neighborhood of the unit element). In other words, if I take a frame in a Lie algebra and interpret it in two ways, as a frame of left invariant vector fields and a frame in right invariant vector fields, then I can take a matrix of real analytic functions on a Lie group and multiply the frame of left invariant vector fields with this matrix to get the correspoding frame of right invariant vector fields. I use in my current research some computations involving this matrix function. Does anybody know of any reference in literature which does any computations involving this matrix valued function on $G$ ?

- Discussion Type
- discussion topicspectral curve
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 10th 2012

New entry spectral curve.

- Discussion Type
- discussion topicToda Bracket and Massey Product
- Category Latest Changes
- Started by jcmckeown
- Comments 5
- Last comment by zskoda
- Last Active Nov 9th 2012

A stub Massey product and a longer Toda bracket (still plenty gaps of reference, many many unlinked words). No promises w.r.t. spellings.

I now see I’ve missed the convention for capitalization. Will fix that now… done.

Cheers

- Discussion Type
- discussion topicinterval object
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Stephan A Spahn
- Last Active Nov 8th 2012

I am hereby moving the following old Discussion box from interval object to here

Urs Schreiber: this is really old discussion by now. We might want to start putting dates on discussions. In principle it can be seen from the entry history, but readers glancing at this here hardly will. Maybe discussions like this here are better had at the forum after all.

We had originally started discussing the notion of interval objects at homotopy but then moved it to this entry here. The above entry grew out of the following discussion we had, together with discussion at Trimble n-category.

*Urs:*Let me chat a bit about what I am looking for here. It seems very useful to have a good notion of what it means in a context like a closed category of fibrant objects to say that*path objects are compatibly corepresented*.By this should be meant: there exists an object $I$ such that

for $B$ any other object, $[I,B]$ is a path object;

and such that $I$ has some structure and property which makes it “nice”.

In something I am thinking about the main point of $I$ being

*nice*is that it can model compositon: it must be possible to put two intervals end-to-end and get an interval of twice the length. In some private notes here I suggest that:a “category with interval object” should be

with a compatible structure of a category of fibrant objects

and equipped with an

**internal co-categoy**on $\sigma, \tau : pt \to I$ for $I$ the*interval object*;such that $I$ co-represents path objects, in that for all objects $B$, $[I,B]$ is a path object for $B$.

I think there are a bunch of obvious examples: all familiar models of higher groupoids (Kan complexes, $\omega$-groupoids etc.) with the interval object being the obvious cellular interval $\{a \stackrel{\simeq}{\to} c\}$.

I also describe one class of applications which I think this is needed/useful for: recall how Kenneth Brown in section 4 of his article on category of fibrant objects (see theorems recalled there and reference given there) describes fiber bundles in the abstract homotopy theory of a

*pointed*category of fibrant objects. This is pretty restrictive. In order to describe things like $\infty$-vector bundles in an context of enriched homotopy theory one must drop this assumption of the ambient category being pointed. The structure of it being a category with an interval object is just the necessary extra structure to still allow to talk of (principal and associated) fiber bundles in abstract homotopy theory. It seems.Comments are very welcome.

*Todd*: The original “Trimblean” definition for weak $n$-categories (I called them “flabby” $n$-categories) crucially used the fact that in a nice category $Top$, we have a highly nontrivial $Top$-operad where the components have the form $\hom_{Top}(I, I^{\vee n})$, where $X \vee Y$ here denotes the cospan composite of two bipointed spaces (each seen as a cospan from the one-point space to itself), and the hom here is the internal hom between cospans.My comment is that the only thing that stops one from generalizing this to general (monoidal closed) model categories is that “usually” $I$ doesn’t seem to be “nice” in your sense here, and so one doesn’t get an interesting (nontrivial) operad when my machine is applied to the interval object. But I’m generally on the lookout for this sort of thing, and would be very interested in hearing from others if they have interesting examples of this.

to be continued in the next comment

- Discussion Type
- discussion topicsuspension isomorphism
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 7th 2012

- Discussion Type
- discussion topicrepresentable morphism of stacks
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 7th 2012

added to

*representable morphism of stacks*the remark that precisely along representable morphisms of stacks over the category of smooth manifolds (i.e. smooth infinity-groupoids) do we have push-forward in generalized cohomology.But I still need to write out some indication justifying this assertion…

- Discussion Type
- discussion topicelliptic chain complex
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 7th 2012

- Discussion Type
- discussion topicloop space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 7th 2012

I noticed that the text at

*loop space*didn’t point to*smooth loop space*and didn’t make clear that such a variant might even exist. So I have now expanded the Idea-section there a little to give a better picture.

- Discussion Type
- discussion topicdirected object
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 6th 2012

I have received a question on the old entry

*directed object*, so I am looking at that now. First of all I’ll clean it up a bit and move old discussion from there to here:

[begin forwarded discussion]

+–{.query}

*Eric*: I don’t fully “grok” this constructive definition, but I like its flavor. Is it possible to formalize the procedure in a simple catchy phrase? In other words, when you begin with a “category $C$ with interval object $I$”, but whose objects are otherwise undirected (like Top), you construct the “supercategory $d_I C$” with directed $C$-objected (even though no objects in $C$ are directed). I used the term “directed internalization”, but is there a better term?I just think this concept is important and should have some really slick arrow theoretic description and I’m not having any luck coming up with one myself.

=–

[ continued in next post ]

- Discussion Type
- discussion topiccomplex geometry - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 6th 2012

finally expanded the long-existing table of contents

*complex geometry - contents*and included it as a floating TOC in the relevant entries.Do we have more entries that need to go here and which I have forgotten?

- Discussion Type
- discussion topicn-category object in an (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 6th 2012

started a stub

*n-category object in an (infinity,1)-category*, to go in parallel with the existing*category object in an (infinity,1)-category*.For the moment, nothing there yet apart from a brief remark that Theta_n spaces are $n$-categories internal to $\infty Grpd$. I hope to expand this entry later.

- Discussion Type
- discussion topicTheta space
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Nov 6th 2012

did I say that I created Theta space?

This is a really nice model. Rezk claims to have shown to get the homotopy hypothesis right for all (n,r)-categories and for both n and r ranging to . If that holds water, it's quite impressive. It seems the only thing missing then is the - Theta-space of all -Theta spaces. Does anyone know if there is a proposal for that?

It's also interesting how the result is a mix of globular and simplicial shapes. So in what respect does that build on/improve over Joyal's original proposal?

- Discussion Type
- discussion topicConcurrency
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Stephan A Spahn
- Last Active Nov 6th 2012

A query about the new entry on copncurrency theory: Does ‘simultaneously’ make sense if there is no global clock?

If not, then the situation gets a lot more like some models for spacetime and the idea of slices through some evolving state space might be a good model.

- Discussion Type
- discussion topicwww.mfo.de/document/1145/OWR_2011_52.pdf
- Category Latest Changes
- Started by FinnLawler
- Comments 3
- Last comment by TobyBartels
- Last Active Nov 6th 2012

Someone, apparently in Berlin, has created a page called www.mfo.de/document/1145/OWR_2011_52.pdf, with just that text (and ’My First Slide’) in the body. The URL points to a report on a logic workshop at Oberwolfach around this time last year. It’s not spam, but what should we do with it?

- Discussion Type
- discussion topicDeletion of a line of text
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by TobyBartels
- Last Active Nov 5th 2012

Someone signing themselves as ‘Joker? at November 3, 2012 08:05:13 from 93.129.88.58’ deleted two lines from sheaf and topos theory. There seemed no reason for this, so I have rolled back to the previous version.

- Discussion Type
- discussion topicstring^c structure
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 5th 2012

added a bit more to

*string^c structure*, but it’s still stubby

- Discussion Type
- discussion topicModal logics
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by Tim_Porter
- Last Active Nov 5th 2012

The recent changes to the various modal logic pages have changed the emphasis from the ’many agent’ versions $S4(m)$.etc. to a type theoretic one. That would be okay but in so doing they have become a bit garbled so they refer to K(m) but then just describe $K$ itself. I am wondering what is planned for these. I originally wrote them with the aim of increasing the nPOV side of the Computer Science entries and to have some brief introduction to modal logs, what should they become?

- Discussion Type
- discussion topicreflective subcategory
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Mike Shulman
- Last Active Nov 3rd 2012

October 24, 2012 09:26:08 by Anonymous Coward (99.133.144.164) has added a comment questioning the validity of a sentence at reflective subcategory.

- Discussion Type
- discussion topicsum
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Mike Shulman
- Last Active Nov 3rd 2012

wanted to be able to say

*sum*and have a pointer to somewhere.

- Discussion Type
- discussion topiclexicographic order, compactification
- Category Latest Changes
- Started by Todd_Trimble
- Comments 13
- Last comment by TobyBartels
- Last Active Nov 2nd 2012

I made starts on lexicographic order and on compactification. Lexicographic order was defined only for products of well-ordered families of linear orders (probably the most common type of application).

I’m not very happy with the opening of compactification.

- Discussion Type
- discussion topicprojection
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by TobyBartels
- Last Active Nov 2nd 2012

I edited the old entry

*projection*a little.There is no real systematics in common use of “projection” as opposed to “projector”, but I think the following makes good sense:

a

*projection*is a canonical map out of a product;a

*projector*is an idempotent in a suitably abelian category

and then the relation is:

*A projector is a projection followed by a subobject inclusion.*That’s how I have now put it in the entry.

- Discussion Type
- discussion topicSchubert cell, intersection theory, enumerative geometry
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 2nd 2012

New entry enumerative geometry. New stubs Schubert calculus, intersection theory.

By the way (Andrew); the title of this nForum post is not seen but truncated. This happens because of some other stuff is placed into the corner in the same line. It says unimportant info “Bottom of Page” preceded by long space between the truncated title and this info ad. I think it is more important that the titles be spelled entirely.

- Discussion Type
- discussion topic5d supergravity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 2nd 2012

stub for

*5-dimensional supergravity*(for the moment I just need the record the reference)