Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012

    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)(x,y) (or [x,y][x,y] as is now more common) for a hom-set. But personally, I like (fg)(f \rightarrow g) (or (fg)(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 ff to gg. —Toby Bartels

    Mike: Perhaps. I never write (x,y)(x,y) for a hom-set, only A(x,y)A(x,y) or hom A(x,y)hom_A(x,y) where AA is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen [x,y][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)E(f,g) or hom E(f,g)hom_E(f,g) if you are considering it as a discrete fibration from AA to BB. 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)(f/g) as less visually distracting, and evidently a generalization of the common notation C/xC/x for a slice category.

    Toby: Well, I never stick ‘EE’ 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)(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)(f,g) in a category is that we have f:XAf:X\to A and g:XBg:X\to B and we’re talking about the pair (f,g):XA×B(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)(f/g/h) for a double comma object makes me like (fgh)(f \to g \to h) even more!

    Mike: I’d rather avoid using \to in the name of an object; talking about projections p:(fg)Ap:(f\to g)\to A looks a good deal more confusing to me than p:(f/g)Ap:(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:ABf, g: A \to B, then fgf \to g ought to be the set of transformations between them. (Or fgf \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)(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)(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 ff and gg 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 CC to DD, that are hom-sets. Finally, I don’t think the notation (f,g)(f,g) scales well to double comma objects; we could write (f,g,h)(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[CfEgD]M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]. Maybe comma[CfEgD]comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]? Lengthy, but at least unambiguous. Or maybe fE I g{}_f {E^I}_g?

    Zoran Skoda: (f/g)(f/g) or (fg)(f\downarrow g) are the only two standard notations nowdays, I think the original (f,g)(f,g) which was done for typographical reasons in archaic period is abandonded by the LaTeX era. (f/g)(f/g) is more popular among practical mathematicians, and special cases, like when g=id Dg = id_D) and (fg)(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)(f/g) then.

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

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

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

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

    =–

    • CommentRowNumber2.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 10th 2017
    • (edited Jun 10th 2017)

    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×d 2d_1\times d_2 which appears not to make sense in that section. Changed.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2017
    • (edited Jun 10th 2017)

    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

    (f/g) E I (FF(a))×(FF(b)) C×D f×g E×E \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

    (f/g) E I (FF(a))×(FF(b)) C×D f×g E×E \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 }
    • CommentRowNumber4.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 12th 2017

    Thanks for the follow-up and explanation of mathrlap.

    There is a related issue concerning Cat. Unsure whether to discuss this here (since it is about a change to Cat. Seems more systematic to open a new discussion with title “Cat”. Will therefore do this.

    • CommentRowNumber5.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 2nd 2017
    • (edited Jul 2nd 2017)

    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 iP_i (for which there apparently is no usual name),

    • the so-called projections H iH_i which are shipped with the comma object point of view.

    It seems natural to ask:

      • how do H iH_i and P iP_i relate to one another? Are these isomorphic functors?

    It seems not usual to mention this question; I have never seen it in the literature.

    It seems useful to discuss

      • whether the nLab article on comma category should briefly discuss and compare these two “canonical” functors from a comma category to the two source categories.

    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.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 2nd 2017

    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.

    • CommentRowNumber7.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 3rd 2017
    • (edited Jul 3rd 2017)

    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,

    • I made what in retrospect seems not a radical decision at all (rather a careful one): to not mention "lax" at all in this article, and to not attempt to define the right notion of 2-categorical limit within comma category, but rather to refer interested readers to 2-limit.

    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.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 3rd 2017

    (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.)

    • CommentRowNumber9.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 3rd 2017
    • (edited Jul 3rd 2017)

    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

    • given the “in components” definition of (f/g), write a proof that the category so defined has the universal property of the comma object.

    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.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 3rd 2017

    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.

    • CommentRowNumber11.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 4th 2017
    • (edited Jul 4th 2017)

    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.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 6th 2017

    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.

  1. A small inclusion in the examples section to illustrate the correspondence between natural transformations between two functors and nice functors from the domain category to the comma category of those two functors.

    T

    diff, v45, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2018

    added hyperlinks to that anonymously-added paragraph .

    diff, v46, current

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJun 28th 2018

    Noted that that example is a special case of the universal property of a comma object.

    diff, v47, current

    • CommentRowNumber16.
    • CommentAuthorJohn Baez
    • CommentTimeOct 13th 2018

    Added information about completeness and cocompleteness of comma categories. The results in Borceux are not as strong as they should be.

    diff, v51, current

  2. I think the conditions for completeness of the comma category were stated incorrectly. Please correct me if I’m wrong.

    diff, v56, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)