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    • Created page. Will fill out more later.

      v1, current

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

    • 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.

      =–

    • Created a stub for nuclear adjunctions.

      v1, current

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

      v1, current

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

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • 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.

    • Discuss how to describe subcategories as displayed categories.

      diff, v36, current

    • fixed the statement of Example 5.2 (this example) by restricting it to 𝒞=sSet\mathcal{C} = sSet

      diff, v36, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • 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 XX.

      diff, v30, current

    • 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.

    • 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!

    • brief category:people-entry for hyperlinking references

      v1, current

    • Clarify that the impredicative definition only quantifies over truth values.

      diff, v18, current

    • 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.

    • For now creating page, content to be added soon.

      v1, current

    • added to G2 the definition of G 2G_2 as the subgroup of GL(7)GL(7) that preserves the associative 3-form.

    • New article mostly clarifying definitions of ’semidefinite integral’ vs ’indefinite integral’ vs ’antiderivative’.

      v1, current

    • I want this entry to have an actual section on construction of (compact) Examples, so I started one (here). But so far there is nothing in there apart from pointers to the original articles by Joyce and Kovalev, and a graphics illustrating Kovalev’s twisted connected sums.

      diff, v51, current

    • For now creating page, content to be added soon.

      v1, current

    • For now creating page, content to be added soon.

      v1, current

    • For now creating page, more to be added soon.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

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

      diff, v7, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • starting something – not done yet

      v1, current

    • example of nominal sets with separated tensor added, see Chapter 3.4 of Pitts monograph Nominal Sets

      Alexander Kurz

      diff, v21, current

    • I create this page to describe precisely the theorem Thomas Fox is known for.

      v1, current

    • 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 =–

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