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- Discussion Type
- discussion topicrelative Langlands program
- Category Latest Changes
- Started by Anton Hilado
- Comments 17
- Last comment by Anton Hilado
- Last Active Feb 12th 2024

- Discussion Type
- discussion topicdistributive law
- Category Latest Changes
- Started by Mike Shulman
- Comments 32
- Last comment by varkor
- Last Active Feb 11th 2024

I added some simpler motivation in terms of the basic example to the beginning of distributive law.

- Discussion Type
- discussion topiccomma category
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by varkor
- Last Active Feb 11th 2024

while bringing some more structure into the section-outline at

*comma category*I noticed the following old discussion there, which hereby I am moving from there to here:

[begin forwarded discussion]

+–{.query} It's a very natural notation, as it generalises the notation $(x,y)$ (or $[x,y]$ as is now more common) for a hom-set. But personally, I like $(f \rightarrow g)$ (or $(f \searrow g)$ if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from $f$ to $g$. —Toby Bartels

Mike: Perhaps. I never write $(x,y)$ for a hom-set, only $A(x,y)$ or $hom_A(x,y)$ where $A$ is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen $[x,y]$ for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.

I would be okay with calling the comma category (or more generally the comma object) $E(f,g)$ or $hom_E(f,g)$

*if*you are considering it as a discrete fibration from $A$ to $B$. But if you are considering it as a*category*in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer $(f/g)$ as less visually distracting, and evidently a generalization of the common notation $C/x$ for a slice category.*Toby*: Well, I never stick ‘$E$’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.*Mike*: The main reason I don’t like unadorned $(f,g)$ for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see $(f,g)$ in a category is that we have $f:X\to A$ and $g:X\to B$ and we’re talking about the pair $(f,g):X\to A\times B$ — surely also a natural generalization of the*very*well-established notation for ordered pairs.*Toby*: The notation $(f/g/h)$ for a double comma object makes me like $(f \to g \to h)$ even more!*Mike*: I’d rather avoid using $\to$ in the name of an object; talking about projections $p:(f\to g)\to A$ looks a good deal more confusing to me than $p:(f/g)\to A$.*Toby*: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If $f, g: A \to B$, then $f \to g$ ought to be the set of transformations between them. (Or $f \Rightarrow g$, but you can't keep that decoration up.)Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation $(f,g)$ is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation $(f,g)$ for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of $f$ and $g$ are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the

*fibers*of the comma category, considered as a fibration from $C$ to $D$, that are hom-sets. Finally, I don’t think the notation $(f,g)$ scales well to double comma objects; we could write $(f,g,h)$ but it is now even less like a hom-set.Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use $M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$. Maybe $comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$? Lengthy, but at least unambiguous. Or maybe ${}_f {E^I}_g$?

Zoran Skoda: $(f/g)$ or $(f\downarrow g)$ are the only two standard notations nowdays, I think the original $(f,g)$ which was done for typographical reasons in archaic period is abandonded by the LaTeX era. $(f/g)$ is more popular among practical mathematicians, and special cases, like when $g = id_D$) and $(f\downarrow g)$ among category experts…other possibilities for notation should be avoided I think.

Urs: sounds good. I’ll try to stick to $(f/g)$ then.

Mike: There are many category theorists who write $(f/g)$, including (in my experience) most Australians. I prefer $(f/g)$ myself, although I occasionally write $(f\downarrow g)$ if I’m talking to someone who I worry might be confused by $(f/g)$.

Urs: recently in a talk when an over-category appeared as $C/a$ somebody in the audience asked: “What’s that quotient?”. But $(C/a)$ already looks different. And of course the proper $(Id_C/const_a)$ even more so.

Anyway, that just to say: i like $(f/g)$, find it less cumbersome than $(f\downarrow g)$ and apologize for having written $(f,g)$ so often.

*Toby*: I find $(f \downarrow g)$ more self explanatory, but $(f/g)$ is cool. $(f,g)$ was reasonable, but we now have better options.=–

- Discussion Type
- discussion topicnuclear adjunction
- Category Latest Changes
- Started by varkor
- Comments 8
- Last comment by varkor
- Last Active Feb 11th 2024

- Discussion Type
- discussion topicwide pullback
- Category Latest Changes
- Started by Guest
- Comments 5
- Last comment by Madeleine Birchfield
- Last Active Feb 10th 2024

- Discussion Type
- discussion topicdependent pullback type
- Category Latest Changes
- Started by Madeleine Birchfield
- Comments 1
- Last comment by Madeleine Birchfield
- Last Active Feb 10th 2024

Moving the section on wide pullbacks or dependent pullbacks in dependent type theory from wide pullback into its own article

- Discussion Type
- discussion topicdependent pushout type
- Category Latest Changes
- Started by Madeleine Birchfield
- Comments 1
- Last comment by Madeleine Birchfield
- Last Active Feb 10th 2024

Moving the section on wide pushouts or dependent pushouts in dependent type theory from wide pullback into its own article

- Discussion Type
- discussion topicD-brane charge quantization in topological K-theory -- references
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Feb 10th 2024

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.

- Discussion Type
- discussion topicLuis Javier Hernández-Paricio > history
- Category Latest Changes
- Started by varkor
- Comments 2
- Last comment by Urs
- Last Active Feb 10th 2024

Presumably this should be merged with Luis Javier Hernández Paricio?

- Discussion Type
- discussion topicCtirad Klimčik > history
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by Urs
- Last Active Feb 10th 2024

Person stub required at moment map.

- Discussion Type
- discussion topicBruno Valette > history
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by Urs
- Last Active Feb 10th 2024

- Discussion Type
- discussion topicErik Van Erp > history
- Category Latest Changes
- Started by varkor
- Comments 2
- Last comment by Urs
- Last Active Feb 10th 2024

Presumably this should be merged with Erik van Erp?

- Discussion Type
- discussion topicopetopic omega-category
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nonemenon
- Last Active Feb 10th 2024

for completeness (prompted by opetopic type theory) I started an entry

*opetopic omega-category*.For me presently this just serves to purpose to record Thorsten Palm’s definition of opetopic omega-category, as I understand it from what Eric Finster tells me.

For the definitions by Baez-Dolan and by Makkai the entry presently only contains placeholders, please feel invited to fill in detail.

All these definitions consider opetopic sets. The difference is in which structure and property is put on that. The original definition of universal cells is somewhat involved, as far as I see. Palm’s definition is of a nice straightforward homotopy-theoretic flavor. It seems plausible that this definition satisfies the homotopy hypothesis, but I don’t know if anyone looked into it.

Accoring to Eric Finster, Palm showed that his definition is a special case of Makkai’s, but the converse remains open.

- Discussion Type
- discussion topicDold-Kan correspondence
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by nLab edit announcer
- Last Active Feb 10th 2024

added reference to dendroidal version of Dold-Kan correspondence

- Discussion Type
- discussion topicsubcategory
- Category Latest Changes
- Started by maxsnew
- Comments 1
- Last comment by maxsnew
- Last Active Feb 9th 2024

- Discussion Type
- discussion topicReedy model structure
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by anuyts
- Last Active Feb 9th 2024

- Discussion Type
- discussion topicYang-Mills theory
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Feb 9th 2024

added these two pointers:

Karen Uhlenbeck, notes by Laura Fredrickson,

*Equations of Gauge Theory*, lecture at Temple University, 2012 (pdf)Simon Donaldson,

*Mathematical uses of gauge theory*(pdf)

(if anyone has the date or other data for the second one, let’s add it)

- Discussion Type
- discussion topicP. K. Mitter
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 9th 2024

- Discussion Type
- discussion topicGerhard Mack
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 9th 2024

- Discussion Type
- discussion topicnon-perturbative quantum field theory
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 9th 2024

added doi-link to

- Franco Strocchi,
*An Introduction to Non-Perturbative Foundations of Quantum Field Theory*, Oxford University Press (2013) [doi:10.1093/acprof:oso/9780199671571.001.0001]

- Franco Strocchi,

- Discussion Type
- discussion topicscheme
- Category Latest Changes
- Started by Richard Williamson
- Comments 20
- Last comment by Dmitri Pavlov
- Last Active Feb 8th 2024

I found the definition of a scheme to be slightly unclear/insufficiently precise at one point, so I have tweaked things slightly, and added more details. Indeed, it is quite common to find a formulation similar to ’every point has an open neighbourhood isomorphic to an affine scheme’, whereas I think it important to be clear that one does not have the freedom to choose the sheaf of rings on the local neighbourhood, it must be the restriction of the structure sheaf on $X$.

- Discussion Type
- discussion topicordered field
- Category Latest Changes
- Started by Madeleine Birchfield
- Comments 19
- Last comment by Madeleine Birchfield
- Last Active Feb 8th 2024

Auke Booij’s thesis Analysis in univalent type theory as well as the HoTT book explicitly defines an ordered field to have an lattice structure on the underlying commutative ring, which is different from the definition of an ordered field in the nlab article, where such a condition is missing. (by lattice I mean unbounded lattice, or what some people call pseudolattices)

However, there are no references in the current nlab article on ordered fields showing that an ordered field doesn’t have a lattice structure in constructive mathematics. The basic definition lacking a lattice structure was already written in 2010 in the first revision of the article by Toby Bartels, and the other editors of the article, Todd Trimble and a few anonymous editors from earlier this year, all accepted the basic definition provided by Toby Bartels, since it hasn’t been modified since the first revision. So if they are still around I would like them or somebody else to provide references from the mathematical literature justifying that ordered fields do not necessarily have a lattice structure, or prove that every ordered field as currently defined has a compatible lattice structure. Otherwise I’ll insert the lattice structure into the definition.

- Discussion Type
- discussion topicreal projective space
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Feb 8th 2024

added a quick note on the CW-structure on real projective space: here.

- Discussion Type
- discussion topicmonoidal category
- Category Latest Changes
- Started by Urs
- Comments 107
- Last comment by Samuel Adrian Antz
- Last Active Feb 8th 2024

Todd,

when you see this here and have a minute, would you mind having a look at

*monoidal category*to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?Thanks!

- Discussion Type
- discussion topiccomplex projective space
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Samuel Adrian Antz
- Last Active Feb 8th 2024

I have split off

*complex projective space*from*projective space*and added some basic facts about its cohomology.

- Discussion Type
- discussion topicSergey Yuzvinsky
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Samuel Adrian Antz
- Last Active Feb 8th 2024

- Discussion Type
- discussion topicJean-Marc Cordier
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Feb 8th 2024

- Discussion Type
- discussion topicnonabelian bundle 2-gerbe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 8th 2024

rediscovered this ancient entry. Added publication data for:

- Branislav Jurčo,
*Nonabelian bundle 2-gerbes*, International Journal of Geometric Methods in Modern Physics**08**01 (2011) 49-78 [arXiv:0911.1552, doi:10.1142/S0219887811004963]

that has become available

*meanwhile*:-)also added pointer to:

- Paolo Aschieri, Branislav Jurčo,
*Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory*, JHEP 0410:068 (2004) [arXiv:hep-th/0409200, doi:10.1088/1126-6708/2004/10/068]

- Branislav Jurčo,

- Discussion Type
- discussion topicpositive element
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 8th 2024

- Discussion Type
- discussion topicsupergravity C-field
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Feb 8th 2024

I am working on the entry supergravity C-field. On the one hand I am in the process of adding in more on the DFM model. On the other I am describing how to reformulate aspects of this in terms of infinity-Chern-Weil theory (this with Domenico Fiorenza and Hisham Sati behind the scenes).

Not done yet, so beware.

- Discussion Type
- discussion topicKalb-Ramond field
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Feb 8th 2024

expanded and polished Kalb-Ramond field. In particular I added more references.

- Discussion Type
- discussion topicKolmogorov extension theorem
- Category Latest Changes
- Started by PaoloPerrone
- Comments 2
- Last comment by PaoloPerrone
- Last Active Feb 8th 2024

- Discussion Type
- discussion topicgauge potential
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 8th 2024

added references and cross-linked with

*electromagnetic potential*(these two entries probably ought to be merged)

- Discussion Type
- discussion topicG2
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Feb 8th 2024

added to

*G2*the definition of $G_2$ as the subgroup of $GL(7)$ that preserves the associative 3-form.

- Discussion Type
- discussion topicdistributive monoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 10
- Last comment by nLab edit announcer
- Last Active Feb 8th 2024

- Discussion Type
- discussion topicMike Shulman
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Mike Shulman
- Last Active Feb 8th 2024

Added

- Mike Shulman,
*Semantics of multimodal adjoint type theory*(arXiv:2303.02572)

the first now of ’Selected writings’.

- Mike Shulman,

- Discussion Type
- discussion topicindefinite integral
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Feb 7th 2024

- Discussion Type
- discussion topicfusion category
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by perezl.alonso
- Last Active Feb 7th 2024

briefly added something to fusion category. See also this blog comment.

- Discussion Type
- discussion topicG2 manifold
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by perezl.alonso
- Last Active Feb 7th 2024

- Discussion Type
- discussion topicM-theory on G2-manifolds
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by perezl.alonso
- Last Active Feb 7th 2024

started

*M-theory on G2-manifolds*

- Discussion Type
- discussion topicbunched logic
- Category Latest Changes
- Started by spitters
- Comments 15
- Last comment by nLab edit announcer
- Last Active Feb 7th 2024

- Discussion Type
- discussion topiczero-one law
- Category Latest Changes
- Started by PaoloPerrone
- Comments 3
- Last comment by PaoloPerrone
- Last Active Feb 7th 2024

- Discussion Type
- discussion topicstochastic dependence and independence
- Category Latest Changes
- Started by PaoloPerrone
- Comments 1
- Last comment by PaoloPerrone
- Last Active Feb 7th 2024

- Discussion Type
- discussion topictopologically twisted D=4 super Yang-Mills theory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by perezl.alonso
- Last Active Feb 7th 2024

started

*topologically twisted D=4 super Yang-Mills theory*, in order to finally write a reply to that MO question we were talking about. But am being interrupted now…

- Discussion Type
- discussion topicde Finetti's theorem
- Category Latest Changes
- Started by PaoloPerrone
- Comments 2
- Last comment by Urs
- Last Active Feb 7th 2024

- Discussion Type
- discussion topicBayesian inversion
- Category Latest Changes
- Started by PaoloPerrone
- Comments 5
- Last comment by PaoloPerrone
- Last Active Feb 7th 2024

- Discussion Type
- discussion topiccoherence theorem for monoidal categories
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by varkor
- Last Active Feb 7th 2024

added an Idea-section to coherence theorem for monoidal categories just with the evident link-backs and only such as to provide a minimum of an opening of the entry

- Discussion Type
- discussion topicHank Chen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 7th 2024

- Discussion Type
- discussion topicDirac monopole
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by Urs
- Last Active Feb 7th 2024

- T.T. Wu, C. N. Yang,
*Dirac monopole without strings: monopole harmonics*, Nuclear Physics B107:3 (1976) 365–380

- T.T. Wu, C. N. Yang,

- Discussion Type
- discussion topicIoannis Markakis
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 7th 2024

- Discussion Type
- discussion topicChristopher J. Dean
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 7th 2024

- Discussion Type
- discussion topiccomputad
- Category Latest Changes
- Started by Mike Shulman
- Comments 24
- Last comment by Urs
- Last Active Feb 7th 2024

Inspired by the discussion at directed n-graph and finite category, added some examples and further explanation to computad.

- Discussion Type
- discussion topicp-adic AdS/CFT correspondence
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Feb 7th 2024

- Discussion Type
- discussion topiccoreader comonad
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by ncfavier
- Last Active Feb 6th 2024

started a minimum at

*writer comonad*

- Discussion Type
- discussion topicsemicartesian monoidal category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 14
- Last comment by varkor
- Last Active Feb 6th 2024

- Discussion Type
- discussion topicFox theorem
- Category Latest Changes
- Started by J-B Vienney
- Comments 5
- Last comment by varkor
- Last Active Feb 6th 2024

- Discussion Type
- discussion topicdouble functor
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by m
- Last Active Feb 6th 2024

- Discussion Type
- discussion topicPlatonism
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Feb 6th 2024

- Discussion Type
- discussion topicnice category of spaces
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by Urs
- Last Active Feb 6th 2024

For some time now I’ve been bothered by an implicit redundancy spanned by the articles nice category of spaces and convenient category of topological spaces. I would like the latter to have a more precise meaning and the former to be something more vague and flexible. I have therefore been doing some rewriting at the former. But if anyone disagrees with the edits, please let’s discuss this here.

I have removed a query box:

+– {: .query} I’m not sure that we really want to use the terminology that way, but Ronnie already created that page, so I’m linking these together. —Toby =–

- Discussion Type
- discussion topicdualizable object
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by varkor
- Last Active Feb 6th 2024

edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories