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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.
=–
comma category had an obvious typo in the section “As a fiber product”: the functor from the functor category over the intervalcat to the square of the common codomain was written as $d_1\times d_2$ which appears not to make sense in that section. Changed.
Not so much a typo as a sudden switch to simplicial notation. But it was good to change it, thanks.
(For others, it’s about the diagram here).
I have however re-instantiated the
\mathrlap{ }
clause in the code. This is necessary when labelling arrows (with label on the right) to keep the position of the arrow fixed. Here it produces
$\array{ (f/g) &\to& E^I \\ \downarrow && \downarrow^{\mathrlap{(F\mapsto F(a))\times(F\mapsto F(b))}} \\ C \times D &\stackrel{f \times g}{\to}& E \times E }$instead of
$\array{ (f/g) &\to& E^I \\ \downarrow && \downarrow^{{(F\mapsto F(a))\times(F\mapsto F(b))}} \\ C \times D &\stackrel{f \times g}{\to}& E \times E }$This is a type of question that often occurs to me, and that most of the time I do not dare to ask, a type characterized by
one does not really need the answer for the structures one is currently trying to construct,
one would nevertheless like to know the answer, out of curiosity and “architectual” interest in the theory,
a modicum of thought and unpacking of definition would probably yield an answer but this would slow one down.
Here, I think it may result in a valuable addition to an article, so I risk asking it, although I suspect that the answer is obvious and can be given conceptually.
So here goes (the details given should be more than sufficient in this forum; if not, I will gladly draw a diagram):
Arguably, given the data defining a comma category, there are two “canonical” functors from the comma category to the source-categories of the defining cospan:
the evident forgetful projections $P_i$ (for which there apparently is no usual name),
the so-called projections $H_i$ which are shipped with the comma object point of view.
It seems natural to ask:
It seems not usual to mention this question; I have never seen it in the literature.
It seems useful to discuss
Like I mentioned above, this should be straightforward to answer, and I did not stop to think about it; my apologies if this is totally obvious and with good reason never discussed.
They are not just isomorphic but equal. However, a more explicit description of how comma categories acquire the universal property of comma objects would not be amiss, either at comma category or comma object.
Thanks for the answer. In comma category I made what I though were easy and improving changes.
Thinking about it, I found the former presenation a bit sawtooth-like. It seems clearer and more linear now.
Notably,
The former version simultaneously mentioned the “lax” usage and warned readers against it. This seems potentially confusing and seems better be confined to the article 2-limit.
(I’ll let Mike answer to the changes announced in #7, but just as a general precept, I get nervous when people who are still learning a subject take it upon themselves to erase material introduced by relatively more experienced people. Usually there are other ways of combatting potential confusions aside from simple erasure.)
but just as a general precept, I get nervous when people who are still learning a subject take it upon themselves to erase material introduced by relatively more experienced people
Okay. Will try to calibrate even more towards conserving as much as possible. And indeed, perhaps everyone except the most experienced should try to be as undestructive as possible.
In this particular case, what made me eventually eliminate “lax” from comma object is the paragraph
Note that lax pullbacks are not the same as comma objects. In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using “lax pullback” to mean “comma object” can be misleading.
in 2-limit. The article comma category, in a sense, did not heed that warning, and had opted for “using “lax pullback” “.
On a technical note, I recognize that strictly speaking I did not carry out what Mike Shulman had been suggesting, namely to perhaps add a proof of
This is still not in comma category, which currently only states that the definitions are equivalent.
Reason was that, partly for lack of time, I thought I make some trivial rearranging changes first (of which the discussion of the “projections” to me clearly seemed to add structure), and make the suggested routine but non-trivial edits perhaps later.
It’s true that in general, even the most experience of us shy away from deleting material written by others, instead trying to incorporate it into new text. I think I support this particular change, however. In fact I would even more strongly be inclined to remove all uses of the word “pullback” to refer to comma objects, even informally.
I would even more strongly be inclined to remove all uses of the word “pullback” to refer to comma objects, even informally.
Tried to do this. Left “pullback” in the section on the 1-categorical pullback. Also left it in in the one sentence on the homotopy pullback.
Changed the label of the section on the 2-limit view, so as not to use “pullback” there.
Harmonised the “foreshadowing” list under “Definition”.
Added “(pb)” in the 1-categorical pullback, which seems to contrast nicely with the explicitly denoted 2-cell in the other diagrams.
In one of the diagrams, an “H_D” is still missing. For lack of time, will add this later, when more care can be applied.
I made a few wording changes, and removed the 2-limit discussion under “Properties” since it is now in the “Definition” section. I also changed my mind about briefly warning the reader against “lax pullback” on this page.
Noted that that example is a special case of the universal property of a comma object.
I have been mulling over what I think would be called the ’lax comma category’ $\mathsf{Cat} // A$ for some small category $A$. This is the same as the usual over-category of diagrams $\mathsf{Cat} \rightarrow A$, except that an arrow between $D_1 \rightarrow A$ and $D_2 \rightarrow A$ consists of an arrow $D_1 \rightarrow D_2$ and a natural transformation from one side of the evident triangle to the other (in the ordinary over-category, one would require that this triangle commutes). One obtains an actual (1-)category in this way.
My first question is: $\mathsf{Cat} // A$ co-complete? I think it is; I think one can use a kind of ’double mapping cylinder’ construction to exhibit what a colimit is, but I think one could also use some argument involving sketches to prove that the category is locally presentable.
My second question is: how far is $\mathsf{Cat} // A$ from being the free co-completion of $A$ (up to equivalence of categories)? Note first that there is a natural ’Yoneda embedding’ $y : A \rightarrow Cat // A$. This is where we use the laxness of $Cat // A$; with the ordinary under-category there are no interesting arrows. If my answer to my first question is correct, then we have co-completeness. So the question boils down to: is every object of $\mathsf{Cat} // A$ (canonically) a colimit of objects in the essential image of $y$? It is not immediately obvious to me that this is false, though I haven’t tried to write down a careful argument.
Hmm, taking $A = 1$, one obtains $\mathsf{Cat}$ and not $\mathsf{Set}$ I think. Perhaps one gets something like the category of presheaves of categories in general, i.e. the $\mathsf{Cat}$-enriched free co-completion.
Hi Richard! I don’t have much to add, but maybe it is worth mentioning that $\mathsf{Cat}//A$ also has a very natural structure as a $2$-category: it is the “lax slice” $\mathsf{Cat}_{/A}$ of $\mathsf{Cat}$ by $A$, given by the bicategorical Grothendieck construction of the $2$-presheaf $\mathsf{Fun}(-,A)\colon\mathsf{Cats}^{\mathsf{op}}\longrightarrow\mathsf{Cats}$.
You can describe it explicitly as follows:
I think the connection with presheaves of categories you mentioned comes from the following alternative description of $\mathcal{C}_{/X}$: using the pseudonatural Yoneda lemma and the fact that the bicategorical Grothendieck construction is a trifunctor, we can equivalently (or rather “biequivalently”, as here we have a biequivalence of $2$-categories) describe $\mathcal{C}_{/X}$ as the bicategorical Grothendieck construction of $\mathsf{PseudoNat}(\mathsf{Fun}(-,-),\mathsf{Fun}(-,A))\cong\mathsf{Fun}(-,A)$. This is the $2$-category whose (now specialising to $\mathcal{C}=\mathsf{Cats}$)
So, in a sense, the lax slice $\mathsf{Cats}_{/A}$ is the full sub-$2$-category of $\mathsf{PseudoFun}(\mathcal{C}^{\mathsf{op}},\mathsf{Cats})$ spanned by the representable pseudopresheaves $\mathsf{Fun}(-,X)\colon\mathcal{C}^{\mathsf{op}}\longrightarrow\mathsf{Cats}$ over $\mathsf{Fun}(-,A)$.
Hi Théo, thank you very much indeed for this! I’ll think some more about it!
Just noticed that the nlab page “lax pullback” redirects to “comma object”, which is a bit of a disaster from a terminological perspective.
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