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The first organism is the body of Earth. The process of nature that leaves Earth in its independency are its solar, lunar and cometary life. This crystal of life is the subject of a meteorological process by which it is fertilized to live. Land and sea blossom in an infinitude of points of puntual temporary life, immesurable phosphorescent points of life in the sea. 

Hegel, 1817 Encyclopedia of the Philosophical Sciences  §338-341

my translation from the German original, cutting some corners to bring out the gist, but otherwise truthful. 

http://ncatlab.org/nlab/show/Science+of+Logic#PN338-341
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David Roberts on Homogeneous bundles and higher geometry. +David Roberts  is visiting the maths institute in Prague. Yesterday he gave a nice talk in the Harmonic Analysis Seminar [0] on string 2-bundle extensions [1] of the canonical principal bundles of the form G -> G/H over Klein geometries [2]. Complete talk notes are in the images below.

This nicely fit into the Higher Cartan Geometry seminar [3]. We'll be discussing some of these constructions later in the seminar. Next session though is about introducing Deligne cohomology [4] and the exact hexagon diagram [5] which it forms 

[0] http://muuk.karlin.mff.cuni.cz/cs/node/13

[1] http://ncatlab.org/nlab/show/differential+string+structure

[2] http://ncatlab.org/nlab/show/Klein%20geometry

[3] http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry#HomogeneousBundlesAndHigherGeometry

[4] http://ncatlab.org/nlab/show/Deligne+cohomology

[5] http://ncatlab.org/nlab/show/differential+cohomology+diagram
 
Content of my talk Homgeneous bundles and higher geometey at the Edouard Čech institute while visiting +Urs Schreiber​. This was intended as a gentle introduction to higher geometry motivated by simple progression of ideas from Klein geometry. I will write notes up and post them very soon. The last few boards contain a sketch of the example from my talk at Herriot-Watt university last June.

(Thanks to Urs for taking the photos during the talk.)
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Higher Cartan Geometry: Motivation. The first session of the higher Cartan geometry seminar tomorrow will informally survey motivation and examples for the theory.

http://ncatlab.org/nlab/show/higher+Cartan+geometry#Motivation

We consider some motivation for higher Cartan geometry from phenomena and open problems visible already in traditional geometry.

1) Pre-quantization of symplectic geometry

2) Higher pre-quantization and Parameterized WZW terms

3) Interlude: Super Cartan geometry

4) Higher pre-quantization and Globalized WZW terms

5) Higher Cartan connections and stacky Cartan geometry

6) Combining it all: Definite higher WZW terms on stacky Cartan geometries

Alternatively, higher Cartan geometry may be motivated intrinsically simply as the result of synthetically formulating Cartan geometry in homotopy type theory. This is the way in which the definition below below proceeds. In the Examples we discuss how this abstract theory indeed serves to inform the motivating phenomena lister here.

1) Pre-quantization of symplectic geometry

While a symplectic manifold structure (X,ω) is an example of an (integrable) G-structure, hence of a Cartan geometry, in many applications symplectic forms ω are to be refined to circle-bundles with connection ∇ (with curvature F∇=ω), a refinement known as prequantization. Notice that while two differential forms on X are either equal or not, two principal connections on X may be different and still equivalent: while there is just a set and hence a homotopy 0-type of symplectic forms on X, there is a groupoid and hence a homotopy 1-type of principal connections on X. It is in this sense that the pair (X,∇) involves higher geometry, namley homotopy n-types for n>0.

In this way the pair (X,∇) is still clearly a geometry of sorts, but not a Cartan geometry. On the other hand, it is still similar enough to be usefully regarded form this perspective:

Just like, by the Darboux theorem, every symplectic manifold has an atlas by charts isomorphic to (R2n,dpi∧dqi), so every prequantum line bundle ∇ on X refining ω is equivalent over this atlas to the U(1)-principal connection given by the globally defined connection for θ≔pi∧dqi.

Moreover, just like the affine symplectic group is the stabilizer group of the local model dpi∧dqi under the canonical Euclidean group-action on R2n, so the homotopy stabilizer group of θ covering this is the extended affine symplectic group which is the semidirect product of the Heisenberg group and the metaplectic group. In this sense metaplectic quantization is a pre-quantized higher analog of symplectic structure.

While one may well reason, evidently, about pre-quantization of symplectic manifolds without a general theory of higher Cartan geometry in hand, this class of examples serves as a first blueprint for what higher Cartan geometry should be like, and points the way to its higher-degree generalizations considerd below.

2) Higher pre-quantization and Parameterized WZW terms

A particularly interesting example of a pre-quantization as above is the Kac-Moody central extension of loop groups of compact semisimple Lie group G (see here). These may be understood as the transgression to loop space of a higher-degree analog of traditional pre-quantization down on G: the canonical left invariant differential 3-form ω3=⟨−,[−,−]⟩ lifts to a circle 2-bundle with connection ∇G. This is also called the WZW gerbe or WZW term, as its volume holonomy serves as the gauge interaction action functional for the Wess-Zumino-Witten sigma model with target space G.

Since the 2-connections on G form a 2-groupoid hence a homotopy 2-type, the pair (G,∇G) may be regarded as being an object in yet a bit higher differential geometry.

Now given a G-principal bundle P→X, then a natural question is whether there is a definite parameterization of ∇G to a 2-form connection on P which restricts fiberwise to ∇G in a suitable sense up to gauge transformation. Such parameterized WZW terms play a key role in heterotic string theory and equivariant elliptic cohomology.

One finds that such definite parameterizations are equivalent to lifts of structure group of the bundle from G to the homotopy stabilizer group of ∇G under the right G-action on itself, and this turns out to be the string 2-group String(G).

While this class of examples is not yet Cartan geometry proper (higher or not) since the bundle here is not a tangent bundle, it contains in it the key aspect of definitie parameterizations of higher pre-quantized forms related to higher G-structures. Such definite parameterizations turn out to be part of genuine examples of higher Cartan geometry, to which we turn below and key ingredients of higher Cartan geometry apply to both cases.

More generally, one considers this situation for WZW terms on coset spaces G/H (the gauged WZW model), and their definite parameterization over G/H-fiber bundles.

3) Interlude: Super-Cartan geometry

Before further motiviating ever higher Cartan geometry, it serves to pause and realize that while passing from manifolds to stacks, we are in particular first of all generalizing to sheaves. So even before going higher in homotopy degree, one may ask how much of Cartan geometry may be formulated in sheaf toposes, first over the site of smooth manifolds itself, which leads to Cartan geoemtry in the generality of smooth spaces, and next over sites other than that of smooth manifolds.

One key example for this is supergeometry. Where a major application of traditonal Cartan geometry is its restriction to orthogonal structures encoding (pseudo-)Riemannian geometry of particular relevance in the theory of gravity, the analogous orthogonal structures in supergeometry serve to set up the theory of supergravity. Indeed, all the traditional literature on supergravity (e.g. (Castellani-D’Auria-Fré 91)) is phrased, more or less explicitly, in terms of Cartan connections for the inclusion of the Lorentz group into the super Poincaré group, this being the formalization of what physicists mean when saying that they pass to “local supersymmetry”.

It so happens that from within such super-Cartan geometry there appear some of the most interesting examples of what should be higher Cartan geometry, hence higher super-Cartan geometry. This we turn to below.

4) Higher pre-quantization and Globalized WZW terms

Given a vector space V equipped with a (constant, i.e. translationally left invariant) differential (p+2)-form

ϕ∈Ωp+2(V)
a natural question to ask is for a V-manifold X (i.e. an n-dimensional manifold if V≃Rn) to carry a differential form

ω∈Ω2(X)
which is a definite form, definite on ϕ, in that its restriction to each tangent space is equal, up to a GL(V)-transformation, to ϕ.

Standard theory of G-structures easily shows that such definite forms correspond to StabGL(V)(ϕ)-structures on X, for StabGL(V)(ϕ) the stabilizer group of ϕ under the canonical GL(V)-action (by pullback of differential forms).

For instance if V=R7 and ϕ∈Ω3(V) is the associative 3-form, then StabGL(V)(ϕ)=G2 is the exceptional Lie group G2 and this yields G2-structures.

But in view of the above discussion one is led to re-state this question for the case that ϕ is refined to a prequantum (p+1)-bundle ∇. A definite globalization of this over a V-manifold X should be a circle (p+1)-connection on X which suitably, up to the relavant higher gauge transformations, restrics locally to ∇.

This problem indeed appears in the formulation of super p-brane sigma models on target super-spacetimes. Here V is a super Minkowski spacetime, ϕ is an exceptional super Lie algebra cocycle of degree (p+2) and the formulation of the Green-Schwarz sigma model requires that it is refined (higher pre-qauntized) to a higher WZW term, a p-form connection. The supergravity equations of motion imply a definite globalization ω of ϕ of a super-spacetime, but to globally define the GS-WZW model one hence needs to lift this globalization to a (p+1)-connection, too (thereby “canceling the classical anomaly” of the model).

These definite globalizations are in particular definite parameterizations, as above, of the restriction of the higher WZW term to the infinitesimal disk-bundle of spacetime, and hence they imply higher G-structure along the above lines.

It is here that developing a theory of higher Cartan geometry has real potential, since, while the globalizations of the forms ϕ have been extensively studied in the literature, the globalization of their pre-quantized refinement to higher WZW-terms has traditionally received almost no attention yet. A brief mentioning of the necessity of considering appears for instance in (Witten 86, p. 17), but traditional tools do get one very far in this question.

More precisely, this is the situation for all those branes in the old brane scan which have no tensor-multiplets on the worldvolume, equivalently those on which no other branes may end (such as the string or the M2-brane, but not the D-branes and not the M5-brane). For more general branes, it turns out that the target space itself is a higher geoemtric space. This leads us to higher Cartan geometry proper. This we turn to now.

5) Higher Cartan connections and Stacky Cartan geometries

A traditonal Cartan connection, being a principal connection satisfying some extra conditions, is locally (on some chart U→X) in particular a Lie algebra valued differential form A∈Ω1(U,g). Following Cartan, this is equivalently a homomorphism of dg-algebras of the form

Ω∙(U)⟵W(g):A
from the Weil algebra of the Lie algebra g to the de Rham complex of U, equivalently a homomorphism of just graded algebras

Ω∙(U)⟵CE(g):A
from the Chevalley-Eilenberg algebra of g. (Requiring this second morphism to also respect the dg-algebra structure, hence the differential, is equivalent to requiring the curvature form FA to vanish, hence to the connection being a flat connection).

In particular for the description of supergravity superspacetimes one considers this for g=Iso(Rd−1,1|N) the super Poincaré Lie algebra of some super Minkowski spacetime Rd−1|N. This serves to encode a Levi-Civita connection as for ordinary gravity modeled by ordinary orthogonal structure Cartan geometry, together with the gravitino field.

In detail, the Chevalley-Eilenberg algebra CE(Iso(R10,1|N=1)) for 11-dimensional Minkowski spacetime turned super via the unique irreducible 32-dimensional spin representation (see here) is freely generated as a graded commutative superalgebra on

elements {ea}11a=1 and {ωab} of degree (1,even);

and elements {ψα}32α=1 of degree (1,odd)
and as a differential graded algebra its differential dCE is determined by the equations

dCEωab=ωab∧ωbc
dCEea=ωab∧eb+i2ψ¯Γaψ.
An algebra homomorphism as above sends these generators to differential forms of the corresponding degree, the vielbein

Ea≔A(ea)∈Ω(1,even)(U),
whe spin connection

Ωab≔A(ωab)∈Ω(1,even)(U)
and the gravitino

Ψα≔A(ψα)∈Ω(1,odd)(U).
But a key aspect of higher dimensional supergravity theories is that their field content necessarily includes, in addition to the graviton and the gravitino, higher differential n-form fields, notably the 2-fom B-field of 10-dimensional type II supergravity and heterotic supergravity as well as the 3-form C-field of 11-dimensional supergravity.

This means that these higher dimensional supergravity theories are not in fact entirely described by super-Cartan geometry. This is to be contrasted with the fact that the very motivation for Cartan geometry, in the original article (Cartan 23), was the mathematical formulation of the theory of gravity (general relativity).

Now a key insight due to (D’Auria-Fré-Regge 80, D’Auria-Fré 82) was that the “tensor multiplet” fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general Chevalley-Eilenberg algebras.

Namely, we may add to the above CE-algabra

a single generator c3 of degree (3,even)
and extend the differential to that by the formula

dCEc3=12ψ¯Γab∧ψ∧ea∧eb.
This still squares to zero due to the remarkable property of 11d super Minkowski spacetime by which 12ψ¯Γab∧ψ∧ea∧eb∈CE4(Iso(10,1|N=1)) is a representative of an exception super-Lie algebra cohomology class. (The collection of all these exceptional classes constitutes what is known as the brane scan).

In the textbook (Castellani-D’Auria-Fré 91) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.

But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of Lie algebras but of strong homotopy Lie algebra, hence of L-infinity algebras, in fact of Lie (p+1)-algebras for (p+1) the degree of the relevant differential form field.

Specifically, me may write the above generalized CE-algebra with the extra degree-3 generator c3 as the CE-algebra CE(m2brane)
of the supergravity Lie 3-algebra m2brane.

Now a morphism

Ω∙(U)←−−CE(m2brane):A
encodes graviton and gravitino fields as above, but in addition it encodes a 3-form

C3≔A(c3)∈Ω(3,even)(U)
whose curvature

G4=dC3+12Ψ¯Γab∧Ψ∧Ea∧Eb
satisfies a modified Bianchi identity, crucial for the theory of 11-dimensional supergravity (D’Auria-Fré 82).

So this collection of differential form data is no longer a Lie algebra valued differential form, it is an L-infinity algebra valued differential form, with values in the supergravity Lie 3-algebra.

The quotient

Rˆ10,1|N=1≔g/h=m2brane/o(R10,1|N=1)
is known as extended super Minkowski spacetime.

The Lie integration of this is a smooth 3-group G which receives a map from the Lorentz group.

This means that a global description of the geometry which (Castellani-D’Auria-Fré 91) discuss locally on charts has to be a higher kind of Cartan geometry which is locally modeled not just on cosets, but on the homotopy quotients of (smooth, supergeometric, …) infinity-groups.

6) Combining it all: Definite higher WZW terms on stacky Cartan geometries

Once such a higher Cartan super-spacetime X as above has been obtained, then we are back to the above question of constructing definite globalizations of WZW terms over it.

Indeed, the super p-brane sigma-models of the D-branes and the M5-brane have WZW terms defined not on plain super Minkowski spacetimes, but on the above extended super Minkowski spacetimes. For instance the WZW term of the M5-brane sigma model is a higher prequantization of the 7-form

ω7≔12ψ¯∧Γa1⋯a5ψ∧ea1∧⋯∧ea5+132ψ¯Γa1a2ψ∧ea1∧ea2∧c3∈CE7(Rˆ10,1|N=1)
on the above extended super-Minkowski spacetime, where c3 is the extra degree-3 generator discussed above.

Under Lie integration this becomes (FSS 13) a degree-7 WZW term defined on a supergeometric 3-group G/H and defining the M5-brane sigma model on a curved supergravity target space means to construct definite globalizations of this over higher Cartan geometries X modeled on this homotopy quotient G/H.

The result (X,LX) is a pair which is still analogous to the symplectic geometries that we started with, but is now in higher geometric homotopy theory in every possible sense.

There is much interesting structure known to be hidden in this pair. One finds (dcct, sections 1.2.11.3 and 1.2.15.3.3) that the higher analog of symplectomorphisms, namely higher quantomorphisms

ϕ:X−→−≃X
η:ϕ∗LX−→−≃L
for the supergeometric 7-group with is the Lie integration of the M-theory Lie algebra of X, witnessing the degree of X being a “BPS state” of 11d supergravity. These BPS states are known to be an immensely rich mathematical topic (e.g. “wall crossing”), but one sees ehre that it is but the local and infinitesimal shadow of a much richer structure: higher isometries in higher super-Cartan geometry.
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+Vít Tuček Thanks for your interest. I'm well aware of what you have quoted from Xu's paper because I spent quite a bit of time tracking down that Dixmier reference. I needed this for Section 4.3 of version 3 of my arXiv preprint (http://arxiv.org/abs/1112.6425) with Armstrong, which was uploaded in March 2012. Xu's own preprint came out in June 2012.

[The reason why my search took so long was because hardly anyone cited Dixmier's book for this. Dixmier himself attributed this result to P. Tauvel, so it could have already been part of folklore at that stage and was considered too well-known (and possibly too trivial) for anyone to go into too much detail about it.]

It's not clear to me whether there are non-parabolic coisotropic subalgebras when g is semisimple.

When g is semisimple, there are no such subalgebras, by Dixmier's lemma. Parabolic subalgebras of semisimple Lie algebras are clearly coisotropic. Dixmier showed that the converse is true, i.e. that coisotropic subalgebras of semisimple Lie algebras must be parabolic. Therefore, in the case of semisimple Lie algebras, the coisotropic subalgebras are precisely the parabolic ones.

What is a coisotropic subalgebra anyway?

You can read Dixmier: the definition is given before the result, which is Lemme 1.1(i) in his book and attributed to P. Tauvel. You can also read Section 4.3 of my arXiv preprint. There is also the definition used by Chen-Stienon-Xu in their work on regular Courant algebroids, which is relevant in this particular context.
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Sombody: "All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings (circles) in a 10 dimensional manifold in order to model particles."

http://www.physicsoverflow.org/27295/space-at-the-planck-scale?show=27351#a27351

No. In perturbative string theory (as well as, in fact, in spectral triples) the input datum is not in general a geometry modeled on manifolds, but is the purely algebraic datum of a 2d SCFT . From this one recovers an "emergent" effective target spacetime at low energies by computing the string perturbation series given by summing up correlators of this abstractly, algebraically defined 2d SCFT and then asking for an ordinary manifold with an ordinary field theory on it which produces the same S-matrix elements at low energy. This may or may not exist! There are plenty of completely non-geometric 2d SCFTs, the famous examples being the Gepner models which model completely non-geometric "phases" of spacetime such as famously the flop transition.

It just so happens that for phenomenological reasons, since one is trying to match to the observed physics well below the Planck scale, there is so much focus on those 2d SCFTs which are constructed geometric as sigma-models from differential geometric input data. But this is human prejudice, not string theory's preference. The moduli space of all 2d SCFTs (the true landscape) has no reason to be dominated by geometric sigma-models. That these are at the focus of attention is because it is easier for us humans to deal with them. One fine day the time will come that mathematical tools have advanced to the point that a genuine analysis of the space of all 2d SCFTs is possible and then we'll be speaking much more about non-geometric backgrounds.

Indeed, for the comparatively simple case of rational 2d CFTs a full mathematical description of the moduli space ("landscape") of all these is already known, by the FRS theorem. And indeed, this classification turns out to be completely algebraic, no differential geometry anywhere. The theorem says that rational 2d CFTs are given essentially by certain Frobenius algebra objects internal to the modular representation categories of rational vertex operator algebras. This does include geometric models such as WZW models, but the general point in this space of rational CFTs has no reason at all to have any relation to smooth manifolds.
All models of space that I know from physics use real or complex manifolds. I was just wondering if it is ... 16 11:45 (UTC), posted by SE-user Q Q
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Ah, I see.
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Announcing "Geometry and Physics 2015: Advances in Perturbation Theory and Feynman Amplitudes", a 6-day event this coming May consisting of a summer school and a conference and covering topics such as amplitudes, exact WKB analysis, cluster algebras and more. Registration will open shortly. For more information, check the webpage below.
Advances in Perturbation Theory and Feynman Amplitudes
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In [1] it was famously observed that the central brane-charge extensions of the super-translation Lie algebras may be understood as the current algebras of the Green-Schwarz super p-branes, the central extension being due to the fact that the corresponding kappa-symmetry super-WZW term is supersymmetric only up to a divergence, so that the Noether theorem for "weak" symmetries applies.

That's great. But following this argument further, there are then gauge-of-gauge symmetries of the divergence term. Have these higher order gauge transformations been discussed anywhere?

[1] José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 ncatlab.org/nlab/show/super Poincare+Lie+algebra#AGIT89
In José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend ... higher order gauge transformations been discussed anywhere?
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Deligne cohomology as you have never seen it before -- unless you read dcct [1]. That's the topic of tomorrow's session of the Higher Cartan Geometry seminar [2]. Detailed lecture notes are now on the nLab [3].

Deligne cohomology is one of those curious instances in the progress of the spirit where the concept springs to the mind of one person [4] in full beauty but the community needs ages to grasp it and instead rediscovers little pieces of the whole story in a tedious lengthy progress. "Bundle gerbes with connection", that's just Deligne cocycles in degree 3 rediscovered over 20 years later, "bundle 2-gerbes" are just Deligne cocycles in degree 4, then finally "holomorphic bundle gerbes" that's actually what Deligne wrote down first! -- and the application of all this to the Beilinson regulators [5] that Deligne was really into still needs to be properly picked up at all.

[1] http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos

[2] http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry

[3] http://ncatlab.org/nlab/show/Deligne+cohomology

[4] http://ncatlab.org/nlab/show/Pierre+Deligne

[5] http://ncatlab.org/nlab/show/higher+regulator
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Maybe to expand on that last statement: the Deligne complex is technically just a minor modification of the de Rham complex. When Artin-Mazur independently considered this 6 years after Deligne in [1] they simply called it the "multiplicative de Rham complex". Their article is another example of the phenomenon of a single paper containing answers that the rest of the field will be being searching for years after the paper is out: they computed what today one might call the Lie differentiation of the smooth moduli stacks of higher gerbes with connection.

[1] http://ncatlab.org/nlab/show/Artin-Mazur%20formal%20group#ArtinMazur77
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Higher extensions of diffeomorphism groups for manifolds equipped with higher topological structure. A note that we are preparing with Domenico Fiorenza and Alessandro Valentino: 

http://ncatlab.org/schreiber/show/Higher+extensions+of+diffeomorphism+groups

We consider higher extensions of diffeomorphism groups and show how these naturally arise as the group stacks of automorphisms of manifolds that are equipped with higher degree topological structures, such as those appearing in topological field theories. Passing to the groups of connected components, we obtain abelian extensions of mapping class groups and investigate when they are central. As a special case, we obtain in a natural way the Z-central extension needed for the anomaly cancellation of 3d Chern-Simons theory.
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Against "spaces" in homotopy theory.  It is popular in some corners to say “space” for “∞-groupoid” /”homotopy type”. Taking the risk of coming across as a hair-splitter trying to waste everybody’s time (and yes, I will, so better stop reading), I hereby voice the opinion that this is a step in the wrong direction. I kow it’s an uphill battle sociologically, but I think it’s for a just cause. While I won’t convince anyone who already sticks to the habit, maybe at least making young students aware of the problem is a just cause for wasting bandwidth with this.

The reason: all along the history of mathematics, the word “space” always referred to a notion of geometric space. All those people in all those centuries who studied vector spaces, Euclidean spaces, affine spaces, Minkowski spaces would have studied the point if you insisted that “space = homotopy type”.

(It is precisely this clash with tradition that for instance in J. Lurie’s Formal Moduli Problems – which sticks to the terminology “space = homotopy type” – requires, below def. 1.2.1, a paragraph of disambiguation.)

The problem begins earlier. The word “space” as used for ∞-groupoids/homotopy types is already a truncation of “topological space” and that truncation by itself is already a step in the wrong direction. Even outside of homotopy theory it is a bad idea to declare that by default “space” is to mean “topological space”. Even if we do, it is a bad idea to conflate homotopy types with the topological spaces that present them. Elsewhere one sees the opinion voiced that choosing a basis of a vector space is “non-gentleman like” behaviour. If so, then conflating topological spaces with their homotopy types is considerably more rude even.

One can see people run into the inconsistency of using “space” to mean “homotopy type” all over the place. For instance everyone accepts the time-honored notion of diffeological spaces. But when I go and say that these are but a special case of “smooth spaces” only to observe that these are naturally embedded into an ambient homotopy theory of smooth infinity-groupoids, I frequently draw flak: They say: “You shouldn’t say ’smooth space’ for what is just a sheaf, for we are likely to confuse it with what you call ’smooth ∞-groupoid’.”

Clearly there is a source of confusion here, but it is not the fault of saying “smooth space” for something that generalizes non-homotopy theoretically what has always been called “diffeological spaces”, and “Euclidean spaces” and has always been a notion of space in geometry.

I could go on and maybe point to Lurie’s Structured Spaces which crucially are not just structured homotopy types. I could point out how in homotopy type theory it has lead to confusion to speak of the homotopy types there as “spaces”, and will lead to more confusion later as the theory develops. And so forth.

In summary: The word “space” in mathematics has always referred to geometric notions of space. Hijacking the word “space” for “∞-groupoid”/”homotopy type” just because topological spaces serve as a presentation for the latter is something that is certainly convenient in, say, the bounded context of a single-focus textbook, but applied more globally to mathematics it goes against the grain of tons of old and good conventions and tradition.

And most importantly: there is no need. The words “∞-groupoid” and “homotopy type” are available and serve their purpose perfectly.

[ this text originally appeared at http://nforum.mathforge.org/discussion/4781/against-spaces-in-homotopy-theory/?Focus=38670#Comment_38670 ]
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Two cents: "web" led me to "network" which can later be reduced to "net".
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There is a curious gap in the supergravity literature concerning the extensions of the susy algebra by those brane charges. I had checked for the literature recently [1] and it seems clear that indeed this had not been discussed properly before:

the seminal article [2] first derives the central super Lie algebra extension of the susy translation Lie algebra by differential forms from computing the Lie algebra of conserved currents of the super p-brane sigma models.

Then, and that's the gap, hands are waved a little and it is argued that some of these differential forms just found are not to count, namely that the exact forms are to be discarded, that the extension is not in fact by differential forms (as just computed) but just by their de Rham cohomology classes.

Physically this is "clear", since these forms are, indeed, currents, whose integral over cycles (under which the exact forms drop out) computes the corresponding charges, and from the physics of branes one expects there to be effects by these net charges. 

But what's the systematic rigorous way to pass from computing a super Lie algebra extension to then discarding some of the extending elements?

I'll tell you what it is: it's homotopy Lie algebras of higher gauge symmetries. The point is that those currents of super p-branes which arise via the generalized Noether theorem from weak symmetries of the kappa-symmetry WZW-action term have themselves higher order gauge transformations between them, by higher order currents (given by lower degree forms). Here two currents are higher gauge equivalent when they differ by the differential of a higher currents. So THAT's why the exact pieces in the extension drop out: while they are present in the super Lie 1-algebra of the symmetries, instead really there is a super Lie n-algebra of higher symmetries, and in there these spurious currents are indeed -- while not on the nose zero -- gauge equivalent to zero.

There is a systematic rigorous way to compute the super Lie n-algebras of higher symmetries of the kappa-symmetry WZW terms (or any other action functional), and it has precisely all these properties. Moreover, we have a theorem that shows that these super Lie n-algebras of higher currents are higher extension by the de Rham cocycle homotopy n-type of the given spacetime. This means that after truncating the super Lie n-algebra down to its 0th Postnikov stage, then it becomes an extension of the plain symmetry algebra by de Rham cohomology. And this is exactly what traditional literature argues for, without, I think, giving a genuine derivation of.

I have now written this out in [3], sections 1.2.11.3 and 1.2.15.3.3

[1] https://plus.google.com/+UrsSchreiber/posts/CYt43HDR5cZ

[2]  José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 ncatlab.org/nlab/show/super Poincare+Lie+algebra#AGIT89

[3] https://dl.dropboxusercontent.com/u/12630719/dcct.pdf
section 1.2.11.3 and 1.2.15.3.3
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Seminar on Higher Cartan Geometry. This semester I am teaching at Charles University of Prague a seminar on theory and applications of higher (stacky) Cartan geometry.

http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry

Classical Cartan geometry is the general theory of differential geometric structures, subsuming (pseudo-)Riemannian geometry, conformal geometry, … symplectic geometry, complex geometry, …, parabolic geometry, etc.

There are however applications which call for structures that are akin to Cartan geometries, but yet a bit richer. For instance:

given a symplectic geometry one asks for its prequantization in which the symplectic form ω is refined to a line bundle with connection;

given a left-invariant (n+1) form ω on some coset space G/H, one asks for its refinement to a WZW term, a cocycle in ordinary differential cohomology, a line n-bundle with connection;

given such a WZW term on G/H, one asks for its globalization over a G/H-Cartan geometry pre-quantizing a definite globalization of its curvature form ω as familiar form special holonomy;

given a map of group stacks H→G (for instance G being the String 2-group corresponding to a Kac-Moody loop group) one asks for an étale stack X locally modeled on the homotopy quotient G/H.

All of these refinements involve higher differential geometry in the sense that they involve geometric homotopy n-types or n-stacks for n>0.

This seminar 1) starts with a self-contained introduction to the elements of higher differential geometry, then 2) presents a theory of higher Cartan geometry in this context, and 3) discusses a collection of examples and applications of this theory.

As a running example, one application which involves all of the above ingredients is super p-brane geometry on higher dimensional supergravity super-spacetimes. Here, higher Cartan geometry serves to properly formulate and then solve problems such as the cancellation of classical anomalies of super p-branes, and the classification of BPS states.

The seminar is based on the material contained in the book-in-preparation

Differential cohomology in a cohesive topos [1]

and the references given there. Concretely, the course notes follow

section 1.2, The geometry of physics [2]

(which will be further expanded as we go along).


[1] http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos
[2] http://ncatlab.org/nlab/show/geometry+of+physics
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I. Fesenko surveys Mochizuki's IU Theory.  Ivan Fesenko, number theorist and artihmetic geometer from Oxford University, has made available a 27 page survey of the main ideas, concepts and objects of the work by Shinichi Mochizuki on interuniversal Teichmüller theory.

http://ncatlab.org/nlab/show/inter-universal+Teichmüller+theory#Fesenko15
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I have to agree with Urs. The stance of some mathematicians and other detractors in this matter reminds me of whiny graduate students who wouldn't put in the work. (I hope I don't come across as being on a moral high horse: it takes one to know one, heh.)

+roux cody By the way, to understand Perelman's proof, it took the maths community almost 4 years, using 3 independent teams, to produce an exegesis of his work. The 4 years is counted from the arXiv postings by Perelman to the time the papers from the 3 teams were published. It has only been 2 years since Mochizuki released his IUT papers, so you may understand why some of us are wondering what the fuss is all about.

And he's not the only one who understands it now. There are others, both in Japan and overseas: see his most recent progress report. The thing now is to have others who did not have direct contact with him to understand his work, so as to provide the truly independent verification that some critics have suggested should take place. Hopefully, Fesenko's survey can help in that.
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