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- Discussion Type
- discussion topicDmitri Faddeev
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 5th 2021

Created:

Dmitri Faddeev (Russian: Дми́трий Константи́нович Фадде́ев) was a Russian mathematician working in Galois theory, group cohomology, and numerical linear algebra. He is the father of Ludwig Faddeev.

He discovered group cohomology independently of Eilenberg and MacLane; the publication of his first paper on this subject was delayed till 1947 because of World War II.

## Related concepts

## Selected writings

On group cohomology:

- Dmitri Faddeev,
*On factor-systems in Abelian groups with operators*. (Russian), Doklady Akad. Nauk SSSR (N. S.) 58, (1947). 361–364.

- Dmitri Faddeev,

- Discussion Type
- discussion topicDijkgraaf-Witten theory
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 5th 2021

Added this (in response to a MathOverflow query):

## Dijkgraaf–Witten theory via Kan extensions

Jeffrey C. Morton,

*Cohomological Twisting of 2-Linearization and Extended TQFT*, arXiv:1003.5603.Section 2 and 3 of Gijs Heuts and Jacob Lurie’s Ambidexterity, in: Topology and Field Theories, doi.

Section 3 of Daniel Freed, Michael Hopkins, Jacob Lurie, Constantin Teleman, Topological quantum field theories from compact Lie groups, arXiv:0905.0731.

- Discussion Type
- discussion topicgroup cohomology
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Dmitri Pavlov
- Last Active Apr 5th 2021

added to group cohomology

in the section structured group cohomology some remarks about how to correctly define Lie group cohomology and topological group cohomology etc. and how not to

in the section Lie group cohiomology a derivation of how from the right oo-categorical definition one finds after some unwinding the correct definition as given in the article by Brylinski cited there.

it's late here and I am now in a bit of a hurry to call it quits, so the proof I give there may need a bit polishing. I'll take care of that later...

- Discussion Type
- discussion topicpointed homotopy type
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Apr 5th 2021

Hi Dmitri, I see you just made “pointed homotopy class” redirect here. But should it not redirect to “homotopy class”, since it’s about maps, not about spaces? I’ll add a line there to highlight the pointed case.

- Discussion Type
- discussion topichomotopy group with coefficients
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Apr 5th 2021

Created:

## Idea

A generalization of homotopy groups.

## Definition

Given a finitely generated abelian group $A$ and $n\ge2$, we set

$\pi_n(X,A)=[P^n(A),X],$where $P^n(A)$ is the $n$th Peterson space of $A$.

## Related concepts

## References

Franklin P. Peterson,

*Generalized Cohomotopy Groups*. American Journal of Mathematics 78:2 (1956), 259–281. doi:10.2307/2372515Joseph A. Neisendorfer,

*Homotopy groups with coefficients*, Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338. doi:10.1007/s11784-010-0020-1.

- Discussion Type
- discussion topicadjoint (infinity,1)-functor theorem
- Category Latest Changes
- Started by Hurkyl
- Comments 1
- Last comment by Hurkyl
- Last Active Apr 5th 2021

- Discussion Type
- discussion topicTyrone Cutler
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 5th 2021

brief

`category:people`

-entry for hyperlinking lecture notes at*Hurewicz fibration*

- Discussion Type
- discussion topicEilenberg-Zilber theorem
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Tim_Porter
- Last Active Apr 5th 2021

created quick entry on Eilenberg-Zilber theorem

(to go with simplicial deRham complex)

but check. I am really in a hurry now and have to leave it in a somewhat stubby state.

- Discussion Type
- discussion topicDieter Puppe
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Apr 5th 2021

- Discussion Type
- discussion topichomotopy class
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 5th 2021

- Discussion Type
- discussion topicFranklin P. Peterson
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

- Discussion Type
- discussion topicPeterson space
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

Created with the following content:

## Definition

Given a finitely generated abelian group $A$ and $n\ge 3$, the $n$th

**Peterson space**$P^n(A)$ of $A$ is the simply connected space whose reduced cohomology groups vanish in dimension $k\ne n$ and the $n$th cohomology group is isomorphic to $A$.## Existence and uniqueness

The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on $A$ and $n$.

There are counterexamples both to existence and uniqueness without these conditions.

For example, the Peterson space does not exist if $A$ is the abelian group of rationals.

## Corepresentation of homotopy groups with coefficients

For all $n\ge2$, we have a canonical isomorphism

$\pi_n(X,A)\cong [P^n(A),X],$where the left side denotes homotopy groups with coefficients and the right side denotes morphisms in the pointed homotopy category.

## Related concepts

## References

- Joseph A. Neisendorfer,
*Homotopy groups with coefficients*, Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338. doi:10.1007/s11784-010-0020-1.

- Joseph A. Neisendorfer,

- Discussion Type
- discussion topicMilnor slide trick
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Richard Williamson
- Last Active Apr 4th 2021

Removed the following discussion to the nForum:

Zoran Škoda: But there is much older and more general theorem of Hurewitz: if one has a map $p:E\to B$ and a numerable covering of $B$ such that the restrictions $p^{-1}(U)\to U$ for every $U$ in the covering is a Hurewicz fibration then $p$ is also a Hurewicz fibration. But the proof is pretty complicated. For example George Whitehead’s

*Elements of homotopy theory*is omitting it (page 33) and Postnikov is proving it (using the equivalent “soft” homotopy lifting property).Todd Trimble: Yes, I am aware of it. You can find a proof in Spanier if you’re interested. I’ll have to check whether the Milnor trick (once I remember all of it) generalizes to Hurewicz’s theorem.

Stephan: I wonder if this trick moreover generalizes (in a homotopy theoretic sense) to categories other that $\Top$; for example to the classical model structure on $Cat$?

- Discussion Type
- discussion topicSamelson product
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 4th 2021

- Discussion Type
- discussion topicJorge Picado
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

- Discussion Type
- discussion topicAleš Pultr
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

- Discussion Type
- discussion topicA. G. Walker
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Apr 4th 2021

Created with the following content:

Arthur Geoffrey Walker was a professor at the University of Liverpool, working in differential geometry, general relativity, and cosmology.

In particular, he is responsible for the Friedmann–Lemaître–Robertson–Walker metric in cosmology.

## Selected writings

On vector fields as derivations:

- W. F. Newns, A. G. Walker,
*Tangent Planes To a Differentiable Manifold*. Journal of the London Mathematical Society s1-31:4 (1956), 400–407 (doi:10.1112/jlms/s1-31.4.400)

- W. F. Newns, A. G. Walker,

- Discussion Type
- discussion topicFRW model
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 4th 2021

some basics at

*FRW model*(in cosmology)

- Discussion Type
- discussion topicvector field
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

Added:

Vector fields can be defined as derivations of the algebra of functions. See the article derivations of smooth functions are vector fields.

- Discussion Type
- discussion topicW. F. Newns
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

- Discussion Type
- discussion topicHans Samelson
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2021

brief

`category:people`

-entry for hyperlinking references at*coset space*and*Samelson product*

- Discussion Type
- discussion topicPaul Mostert
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2021

brief

`category:people`

-entry for hyperlinking references at*closed subgroup*and at*coset space*

- Discussion Type
- discussion topicinductive type
- Category Latest Changes
- Started by Todd_Trimble
- Comments 41
- Last comment by David_Corfield
- Last Active Apr 4th 2021

Based on a private discussion with Mike Shulman, I have added some explanatory material to inductive type. This however should be checked. I have created an opening for someone to add a precise type-theoretic definition, or I may get to this myself if there are no takers soon.

- Discussion Type
- discussion topicbisimplicial set
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Tim_Porter
- Last Active Apr 4th 2021

I have split off bisimplicial set and bisimplicial group from bisimplicial object

- Discussion Type
- discussion topicCartan's homotopy formula
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2021

added pointer to the original reference (as kindly supplied by Dmitri over at

*Lie derivative*):- Élie Cartan,
*Leçons sur les invariants intégraux*(based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).

Also cleaned up some text in this entry here.

- Élie Cartan,

- Discussion Type
- discussion topicderivations of smooth functions are vector fields
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 4th 2021

gave the statement that

*derivations of smooth functions are vector fields*a dedicated entry of its own, in order to be able to convieniently point to it

- Discussion Type
- discussion topicIgor Křiž
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2021

adding accents to the page name, to make Dmitri’s requested links work at

*localic group*

- Discussion Type
- discussion topicLie derivative
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

I am unhappy with Lie derivative. In the previous version it defined the Lie derivative as a secondary notion, using the differential and the Cartan homotopy formula (for which I finally created an entry). I have added a bit mentioning vector fields etc. and a formula using derivatives for forms but this is still not the right thing. Namely, in my understanding the Lie derivative is a

**fundamental notion**and should not be*defined*using other differential operators, but by the “fisherman’s derivative” formula. Second it makes sense not only for differential forms but for any geometric quantities associated to the (co)frame bundle, and in particular to any kind of tensors, not necessarily contravariant or antisymmetrized. For this one has a prerequisite which will require some work in $n$Lab. Namely to a vector field, one associated the flow, not necessarily defined for all times, but for small times. Then for any $t$ one has a diffeomorphism, which is used in the fisherman’s formula. But fisherman’s formula requires the pullback and the pullback is usually defined for forms while for general tensor fields one may need combination of pullbacks and pushforwards. However, for diffeomorphisms, one can define pullback in both cases, and pullback for time $t$ flow corresponds to the pushforward for time $-t$. To define such general pullback it is convenient to work with associated bundles for frame or coframe bundle and define it in the formalism of associated bundles. In the coframe case, this is in Sternberg’s Lectures on differential geometry (what returns me back into great memories of the summer 1987/1988 when I studied that book). So there is much work to do, to add details on those. If somebody has comments or shortcuts to this let me know.However, there is a scientific question here as well: what about when frame bundle is replaced by higher jet bundles, and one takes some higher differential operator for functions and wants to do a similar program – are there nontrivial extensions of Lie derivative business to higher derivatives which does not reduce to the composition of usual Lie derivatives ?

- Discussion Type
- discussion topicChristoffel symbols
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 4th 2021

Added:

Originally due to

- Elwin Bruno Christoffel,
*Über die Transformation der homogenen Differentialausdrücke zweiten Grades*, Journal für die reine und angewandte Mathematik 70 (1869), 46–70. doi:10.1515/crll.1869.70.46, doi:10.1515/9783112389409-003.

- Elwin Bruno Christoffel,

- Discussion Type
- discussion topicaffine connection
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Apr 3rd 2021

Added a reference:

A coordinate-free treatment first appeared in

- Harley Flanders,
*Development of an extended exterior differential calculus*. Transactions of the American Mathematical Society 75:2 (1953), 311–311. doi.

- Harley Flanders,

- Discussion Type
- discussion topicinter-universal Teichmüller theory
- Category Latest Changes
- Started by Urs
- Comments 128
- Last comment by Richard Williamson
- Last Active Apr 3rd 2021

added to

*inter-universal Teichmüller theory*a pointer to the recent note- Yamashita,
*FAQ on ‘Inter-Universality’*(pdf)

(Though after reading I am not sure if that note helps so much.)

- Yamashita,

- Discussion Type
- discussion topictorsor
- Category Latest Changes
- Started by nLab edit announcer
- Comments 15
- Last comment by Richard Williamson
- Last Active Apr 3rd 2021

- Discussion Type
- discussion topicmetric space
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by Dean
- Last Active Apr 3rd 2021

I’ve removed this query box from metric space and incorporated its information into the text:

Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched $\dagger$-categories?

*Toby*: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched $\dagger$-categories.

- Discussion Type
- discussion topicnoncommutative algebraic geometry
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Apr 3rd 2021

Added to

*noncommutative algebraic geometry*a section “Relation to ordinary algberaic geometry” with what is really just a pointer to an article by Reyes:The direct “naive” generalization of Grothendieck-style algebraic geometry via sheaves on a site (Zariski site, etale site etc.) of commutative rings-op to non-commutative rings does not work, for reasons discussed in some detail in (Reyes 12). This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”).

- Discussion Type
- discussion topiclocal trivialization
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 3rd 2021

- Discussion Type
- discussion topicSerre fibration
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Apr 3rd 2021

at

*Serre fibration*I have spelled out the proof that that with $F_x \hookrightarrow X \overset{fib}{\to} Y$ then $\pi_\bullet(F) \to \pi_\bullet(X)\to \pi_{\bullet(Y)}$ is exact in the middle. here.(This is intentionally the low-technology proof using nothing but the definition. )

- Discussion Type
- discussion topicGrothendieck fibration
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Sam Staton
- Last Active Apr 3rd 2021

I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration

I also edited the "Idea"-section at Grothendieck fibration slightly.

That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.

- Discussion Type
- discussion topiccongruence
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by daniel
- Last Active Apr 2nd 2021

touched the formatting in

*congruence*, fixed a typo on the cartesian square, added a basic example

- Discussion Type
- discussion topicCartesian fibration
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Hurkyl
- Last Active Apr 2nd 2021

added more theorems to Cartesian fibration and polished the intro slightly

- Discussion Type
- discussion topicMelvin Rothenberg
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 2nd 2021

brief

`category:people`

-entry for hyperlinking references at*equivariant differential topology*and*equivariant bundle*and maybe elsewhere

- Discussion Type
- discussion topicstratified category
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Apr 2nd 2021

- Discussion Type
- discussion topicmultiset
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Urs
- Last Active Apr 2nd 2021

Removing an old discussion:

Is there a reason that you moved these references up here? We need them especially for the stuff about morphisms below. —Toby

## Discussion

Eric: What would a colimit over an MSet-valued functor $F:A\to MSet$ look like?

*Toby*: That depends on what the morphisms are.Eric: I wonder if there is enough freedom in the definition of morphisms of multisets so that the colimit turns out particularly nice. I’m hoping that it might turn out to be simply the sum of multisets. According to limits and colimits by example the colimit of a Set-valued functor is a quotient of the disjoint union.

*Toby*: I think that you might hope for the coproduct (but not a general colimit) of multisets to be a sum rather than a disjoint union. Actually, you could argue that the sum*is*the proper notion of disjoint union for abstract multisets.

- Discussion Type
- discussion topicYoneda lemma
- Category Latest Changes
- Started by Urs
- Comments 100
- Last comment by David_Corfield
- Last Active Apr 2nd 2021

Todd,

you added to Yoneda lemma the sentence

In brief, the principle is that the identity morphism $id_x: x \to x$ is the universal generalized element of $x$. This simple principle is surprisingly pervasive throughout category theory.

Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element $id_x$ is a tautological statement that does not need or imply the Yoneda lemma, it seems.

- Discussion Type
- discussion topiccategory object in an (infinity,1)-category
- Category Latest Changes
- Started by Mike Shulman
- Comments 10
- Last comment by David_Corfield
- Last Active Apr 2nd 2021

It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.

- Discussion Type
- discussion topicDedekind completion
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Apr 2nd 2021

- Discussion Type
- discussion topicinner product of multisets
- Category Latest Changes
- Started by Eric
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Apr 1st 2021

I created inner product of multisets, which I hope will help make some sense of some speculations over at the discussion of magnitude of metric spaces.

Thank you Toby for your help on my personal wikiweb.

- Discussion Type
- discussion topiccommutative monoid
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active Apr 1st 2021

added missing pointer to

*commutative monoid in a symmetric monoidal category*

- Discussion Type
- discussion topic(infinity,1)-Grothendieck construction
- Category Latest Changes
- Started by Hurkyl
- Comments 2
- Last comment by Hurkyl
- Last Active Apr 1st 2021

- Discussion Type
- discussion topicfunding of the nLab
- Category Latest Changes
- Started by Richard Williamson
- Comments 25
- Last comment by Richard Williamson
- Last Active Apr 1st 2021

Describing the arrangements which have been made for funding of the nLab in collaboration with the Topos Institute. The page, linked to from the home page, is intended to be fairly general; specific requests for donations can be made elsewhere.

- Discussion Type
- discussion topicadjoint (infinity,1)-functor
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Hurkyl
- Last Active Apr 1st 2021

polished and expanded adjoint (infinity,1)-functor

- Discussion Type
- discussion topicadjunction
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Richard Williamson
- Last Active Apr 1st 2021

A comparatively long and technical section “From hom-functors to units and counits” (on adjoint functors) was sitting inside the

*Idea-section*of*adjunction*. It seemed plainly misplaced there, and distracting attention from what should be the content of this entry, as opposed to the entry*adjoint functor*. So I have moved it now to where it seems to belong: inside the Examples-section.

- Discussion Type
- discussion topicDeane Montgomery
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 1st 2021

brief

`category:people`

-entry for hyperlinking references at*slice theorem*

- Discussion Type
- discussion topicSini Karppinen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 1st 2021

brief

`category:people`

-entry for hyperlinking references at*slice theorem*

- Discussion Type
- discussion topicnuclear force
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 1st 2021

I wanted the links to

*weak nuclear force*and*strong nuclear force*in various entries to cease appearing grayish and ugly. So I created a minimal entry*nuclear force*.

- Discussion Type
- discussion topicshort exact sequence of vector bundles
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 1st 2021

- Discussion Type
- discussion topiccategory of filters
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Urs
- Last Active Apr 1st 2021

This is a short article with a definition of the category of filters, now called the category of filters.

I apologise for vandalising by mistake the article on filters, and thanks to Richard Williamson for fixing this. I tried to “edit a current page .. in context on a relevant page” misunderstanding the intructions from HowTo:

How to start a new page

You do this in two steps, the first of which may have already been done:

Create a preliminary link (represented by a question mark) by editing > a current page and putting the name of the new page in double square brackets

Anonymous

- Discussion Type
- discussion topicHowTo
- Category Latest Changes
- Started by Urs
- Comments 45
- Last comment by Urs
- Last Active Apr 1st 2021

Have added to HowTo a description for how to label equations

In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

- Discussion Type
- discussion topicvertical vector field
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Apr 1st 2021

- Discussion Type
- discussion topicalgebraic topology
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Apr 1st 2021

I have tried to give

*algebraic topology*a better Idea-section.

- Discussion Type
- discussion topicD-brane charge quantization in topological K-theory -- references
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 1st 2021

this is a bare list of references, to be

`!include`

-ed into relevant entries (such as*D-brane*,*Dirac charge quantization*and*D-brane charge quantization in K-theory*).In fact, the list is that which has been in each of these entries all along, and it has been a pain to synchronize the parallel lists. So this here now to ease the process.