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.

=–

]]>following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

]]>since this was missing, I created a minimum at *equivalence in a 2-category*

added to *polynomial monad* the article by Batanin-Berger on homotopy theory of algebras over polynomial monads.

At *2-category equipped with proarrows* in the section *As a double category* I have made a little change in the labelling:

there used to be a horizontal arrow labeled “$K$”, but also the ambient 2-category (the one being equipped) is denoted “$K$” and sometimes both symbols, or rather the same symbol with its two different meanings, appeared right next to each other.

So I have relabeled the horizontal arrow now to “$J$”. I tried to take care to do so consistently throughout the paragraph… Hopefully you can agree with this change.

One question: a few months back we chatted vaguely about how equipment data is equivalent to the structure of an internal category in Cat in the sense at internal (infinity,1)-category. Back then I had written a quick note on this at *Segal space - Examples - in 1Grpd*.

I’d like to expand on that. Is there meanwhile anything in this direction in the literature?

]]>Has anyone worked out the universal property of the Cauchy completion of an enriched category?

What I have in mind is a definition that makes sense in any (suitable) framed bicategory, such as $\mathcal{V}$-Prof. Is there a notion of when an object in a framed bicategory has all absolute limits, formulated in terms of something like absolute Kan extensions? If so, is there a definition of Cauchy completion in these terms?

]]>started *co-Kleisli category* with a minimum of content. Even though its formally dual to *Kleisli category*, of course, it may be worthwhile to have a separate entry.

started a stub for *ambidextrous adjunction*, but not much there yet

stub for *compact closed 2-category* to accomodate a pointer to Mike Stay’s recent article

I have taken the liberty of moving one more bit of Mike’s stuff on 2-topos theory from his personal web to the main $n$Lab, namely *core in a 2-category*.

(I am trying to fill what used to be the gray links in the proof at *2-topos – In terms of internal categories*).

I have taken the liberty of moving one more bit of Mike’s stuff on 2-topos theory from his personal web to the main $n$Lab, namely *n-localic 2-topos*.

(I am trying to fill what used to be the gray links in the proof at *2-topos – In terms of intenral categories*).

*sylleptic monoidal 2-category*, *symmetric monoidal 2-category*

… but now I am running out of steam…

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