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- Discussion Type
- discussion topicAlexander Goncharov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicVladimir V. Fock
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicFock-Goncharov coordinates
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicanabelian geometry
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by nLab edit announcer
- Last Active 7 days ago

- Discussion Type
- discussion topicAmar Hadzihasanovic
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by David_Corfield
- Last Active 7 days ago

- Discussion Type
- discussion topicstring diagram
- Category Latest Changes
- Started by nLab edit announcer
- Comments 48
- Last comment by David_Corfield
- Last Active 7 days ago

- Discussion Type
- discussion topicnoncommutative principal bundle
- Category Latest Changes
- Started by perezl.alonso
- Comments 2
- Last comment by perezl.alonso
- Last Active 7 days ago

fixed link for

- Tomasz Brzeziński.
*On synthetic interpretation of quantum principal bundles*, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213

- Tomasz Brzeziński.

- Discussion Type
- discussion topicKK-theory
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by perezl.alonso
- Last Active 7 days ago

added to

*KK-theory*brief remark and reference to relation to stable $\infty$-categories / triangulated categories

- Discussion Type
- discussion topicflux tube
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicVladimir Folomeev
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicVladimir Dzhunushaliev
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicAlexandru Proca
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicProca theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicnonunital algebra
- Category Latest Changes
- Started by perezl.alonso
- Comments 1
- Last comment by perezl.alonso
- Last Active Apr 11th 2024

- Discussion Type
- discussion topicmicrolinear space
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Dmitri Pavlov
- Last Active Apr 11th 2024

created microlinear space

One thing I might be mixed up above:

in the literature I have seen it seems to say that

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

with

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

But shouldn't it be

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

?

- Discussion Type
- discussion topicDay convolution
- Category Latest Changes
- Started by Harry Gindi
- Comments 60
- Last comment by varkor
- Last Active Apr 11th 2024

See Day convolution

I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category $V^{A^{op}}$ and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).

This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.

- Discussion Type
- discussion topicMichel Talagrand
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2024

- Discussion Type
- discussion topicquantum field theory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 11th 2024

wrote an Idea-section at quantum field theory

- Discussion Type
- discussion topiccontinuous functor
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Apr 11th 2024

added pointer to:

- Peter Freyd, Max Kelly,
*Categories of continuous functors*, J. Pure. Appl. Algebra**2**(1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

removed the following ancient query box discussion:

+–{.query} Left I could understand, but right? —Toby

The way I rewrote it explains it. It is unfortunate that the Eilenberg-Watts theorem treated in Bass was using only right adjoint functors so later they dropped word right. – Zoran

Thanks. —Toby =–

- Peter Freyd, Max Kelly,

- Discussion Type
- discussion topicBFSS matrix model
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Apr 11th 2024

- Discussion Type
- discussion topicYoneda lemma
- Category Latest Changes
- Started by Urs
- Comments 107
- Last comment by nLab edit announcer
- Last Active Apr 11th 2024

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 topicSergio Ferrara
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2024

- Discussion Type
- discussion topicEugène Cremmer
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 11th 2024

- Discussion Type
- discussion topicadinkra
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 11th 2024

added pointer to yesterday’s

- Jim Gates, Yangrui Hu, S.-N. Hazel Mak,
*Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, $\mathcal{N} = 1$ Superspace*(arXiv:2002.08502)

- Jim Gates, Yangrui Hu, S.-N. Hazel Mak,

- Discussion Type
- discussion topicfiber integration in K-theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Apr 11th 2024

I am giving

*fiber integration in K-theory*a dedicated entry.One section

*In operator K-theory*used to be a subsection of*fiber integration in generalized cohomology*, and I copied it over.Another section

*In terms of bundles of Fredholm operators*I have now started to write.

- Discussion Type
- discussion topicBTZ black hole
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active Apr 11th 2024

- Discussion Type
- discussion topicquasicoherent sheaf
- Category Latest Changes
- Started by Urs
- Comments 49
- Last comment by perezl.alonso
- Last Active Apr 10th 2024

- following up on our discussion in the thread <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=593&page=1#Item_12">oo-vector bundle (forum)</a> here on the forum, I have now spent a bit of time on expanding the entry quasicoherent sheaf

- I fixed the formulas in the section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheaves">As sheaves on Aff/X</a>. They were a bit rough and typo-ridden in the first version (which likely was my fault, not anyone else's). Since there are 50 variants in the literature to state this, I also pointed to page and verse in a lecture by Goerss where the statement is given explicitly the way it now appears there in the entry

- then I added a new section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheaves">As homs into the stack of modules</a> where I aim to describe in great detail how this definition is equivalent to the even simpler one, where we just say that the category of quasicoherent sheaves on a sheaf is the Hom-category for the stack of modules, classifying the canonical bifibration . This is the statement that my discussion at <a href="http://ncatlab.org/schreiber/show/%E2%88%9E-vector+bundle">oo-vector bundle (schreiber)</a> was secretly based on, which I promised to make more explicit.

- then I added a section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#Higher">Higher/derived quasicoherent sheaves</a>, where I indicate the now obvious oo-categorification discussed in more detail at <a href="http://ncatlab.org/schreiber/show/%E2%88%9E-vector+bundle">oo-vector bundle (schreiber)</a> and point out how this gives the derived QC sheaves used by Ben-Zvi et al as discussed at geometric infinity-function theory

- finally I wrote a fairly detailed <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#Idea">Idea</a> section for quasicoherent sheaves, that previews the content of all these sections.

- Discussion Type
- discussion topicsupermanifold
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Apr 10th 2024

I have added a little bit to supermanifold, mainly the definition as manifolds over superpoints, the statement of the equivalence to the locally-ringed-space definition and references.

- Discussion Type
- discussion topicframe field
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 10th 2024

added pointer to:

- Pietro Fré, §3.2.5 in:
*Gravity, a Geometrical Course*, Volume 1:*Development of the Theory and Basic Physical Applications*, Spinger (2013) [doi:10.1007/978-94-007-5361-7]

here and elsewhere

- Pietro Fré, §3.2.5 in:

- Discussion Type
- discussion topicgeneral relativity
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Apr 10th 2024

I wanted to be able to use the link without it appearing in grey, so I created a stub for general relativity.

- Discussion Type
- discussion topicfiber bundles in physics
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active Apr 10th 2024

Started an entry in “category:motivation” on

*fiber bundles in physics*.(prompted by this Physics.SE question)

- Discussion Type
- discussion topicGamma-set
- Category Latest Changes
- Started by David_Corfield
- Comments 8
- Last comment by Urs
- Last Active Apr 10th 2024

- Discussion Type
- discussion topicMOND
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Urs
- Last Active Apr 10th 2024

recorded some recent surveys of the status of MOND at

*MOND*

- Discussion Type
- discussion topicelliptic genera as partition functions -- references
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Apr 10th 2024

a bare list of references, to be

`!include`

-ed into relevant entries (such as*Witten genus*,*M5-brane elliptic genus*but also inside*elliptic cohomology – references*) – for ease of harmonizing lists of references

- Discussion Type
- discussion topicUrko Reinosa
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 10th 2024

- Discussion Type
- discussion topicconfinement
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by Urs
- Last Active Apr 10th 2024

stub for

*confinement*, but nothing much there yet. Just wanted to record the last references there somewhere.

- Discussion Type
- discussion topicskyrmion
- Category Latest Changes
- Started by Urs
- Comments 60
- Last comment by Urs
- Last Active Apr 10th 2024

- Discussion Type
- discussion topicbar construction
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Dmitri Pavlov
- Last Active Apr 9th 2024

the standard

*bar complex*of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).

(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)

- Discussion Type
- discussion topic11d SuGra from super C-field flux quantization -- section
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Apr 9th 2024

starting a stand-alone Section-entry (to be

`!include`

ed as a section into*D=11 supergravity*and into*D’Auria-Fré formulation of supergravity*)So far it contains lead-in and statement of the result, in mild but suggestive paraphrase of CDF91, §III.8.5.

I am going to spell out at least parts of the proof, with some attention to the prefactors.

- Discussion Type
- discussion topicSo Nakamura
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 9th 2024

- Discussion Type
- discussion topicJonathan Beardsley
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by David_Corfield
- Last Active Apr 9th 2024

changed entry title to full name,

added “category:people” tag,

updated webpage url,

added section “Related nLab entries”, so far with a pointer to

*Grothendieck construction*

- Discussion Type
- discussion topicfield with one element
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by David_Corfield
- Last Active Apr 9th 2024

created field with one element with two useful references

- Discussion Type
- discussion topicself-dual higher gauge theory
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Apr 9th 2024

started self-dual higher gauge theory. Just minimal idea and list of references so far.

- Discussion Type
- discussion topicquark-gluon plasma
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 9th 2024

am adding references, such as this one:

- Francesco Biagazzi, A. l. Cotrone,
*Holography and the quark-gluon plasma*, AIP Conference Proceedings 1492, 307 (2012) (doi:10.1063/1.4763537, slides pdf)

- Francesco Biagazzi, A. l. Cotrone,

- Discussion Type
- discussion topicrigid monoidal category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 8
- Last comment by nLab edit announcer
- Last Active Apr 8th 2024

Added:

### Free rigid monoidal categories

The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor $L$.

Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category $C$ admits a fully faithful strong monoidal functor $C\to L(C)$, where $L(C)$ is a rigid monoidal category.

See Theorems 1 and 2 in Delpeuch \cite{Delpeuch}.

- Antonin Delpeuch,
*Autonomization of monoidal categories*, arXiv, doi.

- Antonin Delpeuch,

- Discussion Type
- discussion topictabulator
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 8
- Last comment by varkor
- Last Active Apr 6th 2024

- Discussion Type
- discussion topicmathematics presented in homotopy type theory
- Category Latest Changes
- Started by David_Corfield
- Comments 39
- Last comment by Ulrik
- Last Active Apr 6th 2024

- Discussion Type
- discussion topicEilenberg-Moore category
- Category Latest Changes
- Started by FinnLawler
- Comments 38
- Last comment by varkor
- Last Active Apr 6th 2024

I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

- Discussion Type
- discussion topicrelative monad
- Category Latest Changes
- Started by mattecapu
- Comments 18
- Last comment by varkor
- Last Active Apr 6th 2024

- Discussion Type
- discussion topicrelative adjoint functor
- Category Latest Changes
- Started by Peter Heinig
- Comments 3
- Last comment by varkor
- Last Active Apr 6th 2024

Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.

- Discussion Type
- discussion topicformal category theory
- Category Latest Changes
- Started by David_Corfield
- Comments 43
- Last comment by varkor
- Last Active Apr 6th 2024

- Discussion Type
- discussion topicEncyclopedia of Mathematical Physics 2nd ed
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by Urs
- Last Active Apr 6th 2024

starting a

`category:reference`

-entry.Just a single item so far, but this entry should incrementally grow as more preprints appear (similar to what we have been doing at

*Handbook of Quantum Gravity*and similar entries).I know that a soft deadline for submissions of at least one of the sections is this December, so I am guessing this is planned to appear in 2024.

- Discussion Type
- discussion topicLie n-groupoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Apr 5th 2024

- Discussion Type
- discussion topiccompactly generated topological space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 90
- Last comment by peter_bonart
- Last Active Apr 5th 2024

I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.

- Discussion Type
- discussion topicdouble dimensional reduction
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Apr 5th 2024

am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.

started a stubby

*double dimensional reduction*in this context and added some first further pointers and references to*M5-brane*, to*D=5 super Yang-Mills theory*and maybe elsewhere.But this still needs more details to be satisfactory, clearly.

- Discussion Type
- discussion topicalgebraic K-theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 50
- Last comment by Urs
- Last Active Apr 5th 2024

I have started on a revision of algebraic K-theory. The old version launched straight into a particular nPOV, which really just summarised the Blumberg et al paper, and did not mention any of the other ideas in the area. At present I have just put in some historical stuff, but given the importance of the subject e.g. in modern C*-algebra the page needs a lot more work.

- Discussion Type
- discussion topicAdS-CFT in condensed matter physics
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active Apr 5th 2024

- Discussion Type
- discussion topicchiral perturbation theory
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Apr 5th 2024

- Discussion Type
- discussion topichyperdoctrine
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Apr 4th 2024

- Discussion Type
- discussion topictangent category
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by nLab edit announcer
- Last Active Apr 4th 2024

created tangent category

in order to have a place where to keep just details on the purely 1-categorical "shadow" of tangent (infinity,1)-category.