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- Discussion Type
- discussion topicMathOverflow
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active May 20th 2020

- Discussion Type
- discussion topicmeasurable locale
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by Dmitri Pavlov
- Last Active May 20th 2020

I wrote about Dmitri Pavlov’s concept of measurable locales.

- Discussion Type
- discussion topiclist of journals publishing category theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 40
- Last comment by Dmitri Pavlov
- Last Active May 20th 2020

- Discussion Type
- discussion topicempty 203
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active May 20th 2020

- Discussion Type
- discussion topicAnne Marie Svane
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 20th 2020

- Discussion Type
- discussion topiccobordism category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 20th 2020

added this pointer on the homotopy groups of the embedded cobordism category:

Marcel Bökstedt, Anne Marie Svane,

*A geometric interpretation of the homotopy groups of the cobordism category*, Algebr. Geom. Topol. 14 (2014) 1649-1676 (arXiv:1208.3370)Marcel Bökstedt, Johan Dupont, Anne Marie Svane,

*Cobordism obstructions to independent vector fields*, Q. J. Math. 66 (2015), no. 1, 13-61 (arXiv:1208.3542)

- Discussion Type
- discussion topicinfinity-groupoid
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 23
- Last comment by Mike Shulman
- Last Active May 19th 2020

- Discussion Type
- discussion topicRadon measure
- Category Latest Changes
- Started by PaoloPerrone
- Comments 20
- Last comment by DavidRoberts
- Last Active May 19th 2020

- Discussion Type
- discussion topicsmall presheaf
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 5
- Last comment by Mike Shulman
- Last Active May 19th 2020

Added a reference.

Can we say exactly what kind of pretopos the category of small presheaves on a category C is?

Is it a ΠW-pretopos, provided that PC is complete?

- Discussion Type
- discussion topicPeter May
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 17
- Last comment by Mike Shulman
- Last Active May 19th 2020

- Discussion Type
- discussion topicopen problem of confinement -- references
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

a bare sub-section with a list of references – to be

`!included`

into relevant entries – mainly at*confinement*and at*mass gap problem*(where this list already used to live)

- Discussion Type
- discussion topiccolor branes and flavor branes
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topicbackreaction
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topicRaman Sundrum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

brief

`category:people`

-entry for hyperlinking references at*Randall-Sundrum model*

- Discussion Type
- discussion topicLisa Randall
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

for hyperlinking references at

*Randall-Sundrum model*and at*probe brane*

- Discussion Type
- discussion topicPeter Ouyang
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

brief

`category:people`

-entry for hyperlinking references at*flavor brane*and at*probe brane*

- Discussion Type
- discussion topicprobe brane
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topicETCC
- Category Latest Changes
- Started by Thomas Holder
- Comments 2
- Last comment by Thomas Holder
- Last Active May 19th 2020

- Discussion Type
- discussion topicMelvin Hochster
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topicHochster duality
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Tim_Porter
- Last Active May 19th 2020

- Discussion Type
- discussion topichypercharge
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topiccomma category
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Matthijs Vákár
- Last Active May 19th 2020

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 topicconfinement
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Urs
- Last Active May 19th 2020

stub for

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

- Discussion Type
- discussion topicchiral perturbation theory
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topicgauge theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active May 19th 2020

- Discussion Type
- discussion topicgeometric theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 34
- Last comment by Thomas Holder
- Last Active May 19th 2020

Added a description of several different approaches to geometric theory.

- Discussion Type
- discussion topicfully normal spaces are equivalently paracompact
- Category Latest Changes
- Started by arsmath
- Comments 2
- Last comment by arsmath
- Last Active May 19th 2020

- Discussion Type
- discussion topiclocalic topos
- Category Latest Changes
- Started by mattecapu
- Comments 5
- Last comment by Thomas Holder
- Last Active May 19th 2020

- Discussion Type
- discussion topicBehrang Noohi
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active May 19th 2020

- Discussion Type
- discussion topicsecond-countable regular spaces are paracompact
- Category Latest Changes
- Started by arsmath
- Comments 1
- Last comment by arsmath
- Last Active May 19th 2020