Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 30th 2009
   • PermaLink
   Author: Andrew Stacey
   Format: Markdown[[Tall-Wraith monoid]] Updated the reference to "The Hunting of the Hopf Ring" since it's now appeared in print.

   Tall-Wraith monoid

   Updated the reference to "The Hunting of the Hopf Ring" since it's now appeared in print.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 26th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI archive here the superlong query box from [[Tall-Wraith monoid]]. > [[Andrew Stacey]]: I changed the $V$ to $\mathcal{V}$ in line with the notation from _The Hunting of the Hopf Ring_. Should there be a lab standard on fonts for categories, objects, functors, and the like? > _Toby_: Based on [this discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=37), ‘$\mathcal{V}$’ is more likely to cause problems for people that haven\'t installed appropriate fonts. (Not that I\'m changing it back, mind you ....) > [[John Baez]]: I hate 'fancy fonts for fancy gizmos', because they mainly serve to make gizmos seem fancy and intimidating. That's why I had changed $\mathcal{V}$ to $V$. I think I'll change it back again, just to annoy Andrew. It's just a category with products, for god's sake! Surely we can't insist on calligraphic font whenever we see one of those: must we say $\mathcal{Set}$? (Yes, I'm being ornery.) > [[Andrew Stacey]]: Okay, okay, I surrender! I'll even change back the ones that you missed. > But I reject the charge of 'fancy fonts for fancy gizmos'. My intention was to use the change of fonts to represent the relationships between things. Thus $a$ is an element of $A$ which is an object of $\mathcal{A}$ which is mapped by the functor $\mathfrak{A}$. It's a bit like using $m,n$ for natural numbers. Yes, we can and often do use $a$ for a natural number, but if I scan a page and see $n$ then I automatically think "natural number" (okay, maybe integer). Since that is reasonably widespread, authors can use it to make their work clearer so that the reader doesn't continually have to go back to the first page to work out whether $A$ was the category or the object. Of course we _can_ write > $$ \forall \delta \forall x \gt 0 \exists y \gt 0 : \forall \epsilon, |\epsilon - \delta| \lt y \implies |f(\epsilon) - f(\delta)| \lt x $$ > but, frankly, wouldn't you rather we didn't? > [[John Baez]]: In lots of ordinary math the strategy of 'bigger fancier fonts for bigger fancier things' works fine: $a$ for an element of the set $A$ in the category $\mathcal{A}$. But the problem is that in $n$-category theory everything is an object in some $n$-category... including every $n$-category, and every functor between them. So, the idea of a clear hierarchy of two or three types of things, with bigger fancier things getting bigger fancier fonts, breaks down. For example, while you might want to say > "Consider the functor $\mathfrak{A}: \mathcal{A} \to \mathcal{A}$..." > you'll often want to continue that sentence with > "... and think of it as an object $\mathfrak{A} \in End(\mathcal{A})$. Now, since $End(\mathcal{A}) \in MonCat$, it makes sense to equip $\mathfrak{A}$ with the structure of a monoid object, and we then call it a **monad**." > This is just a very simple example of the kind of fancy level-shifting that takes place. And this level-shifting is _precisely the style of thinking that makes $n$-category theory so powerful_. > For this reason, at some point I decided Jim Dolan was very wise to put everything in the same font: it encourages mental flexibility. I later impressed Ross Street greatly when I wrote on the blackboard "Consider a bicategory $x$." > Even though I'll be eternally proud of that moment, I think it can be good to use a couple of different fonts in a given discussion to help 'set the types' of the entities in question --- hence the term 'typesetting'. But this sort of distinction is only helpful _locally_: it's not so good to demand _globally_ consistent typesetting. Your big fancy algebraic theory $\mathcal{A}$ is likely to become someone else's puny little object $a$ in $FinProdCat$, and then $FinProdCat$ will become a puny little object in $Doctrines$, and so on.

   I archive here the superlong query box from Tall-Wraith monoid.

   Andrew Stacey: I changed the $V$ to $\mathcal{V}$ in line with the notation from The Hunting of the Hopf Ring. Should there be a lab standard on fonts for categories, objects, functors, and the like?

   Toby: Based on this discussion, ‘$\mathcal{V}$’ is more likely to cause problems for people that haven't installed appropriate fonts. (Not that I'm changing it back, mind you ….)

   John Baez: I hate ’fancy fonts for fancy gizmos’, because they mainly serve to make gizmos seem fancy and intimidating. That’s why I had changed $\mathcal{V}$ to $V$. I think I’ll change it back again, just to annoy Andrew. It’s just a category with products, for god’s sake! Surely we can’t insist on calligraphic font whenever we see one of those: must we say $\mathcal{Set}$? (Yes, I’m being ornery.)

   Andrew Stacey: Okay, okay, I surrender! I’ll even change back the ones that you missed.

   But I reject the charge of ’fancy fonts for fancy gizmos’. My intention was to use the change of fonts to represent the relationships between things. Thus $a$ is an element of $A$ which is an object of $\mathcal{A}$ which is mapped by the functor $\mathfrak{A}$. It’s a bit like using $m,n$ for natural numbers. Yes, we can and often do use $a$ for a natural number, but if I scan a page and see $n$ then I automatically think “natural number” (okay, maybe integer). Since that is reasonably widespread, authors can use it to make their work clearer so that the reader doesn’t continually have to go back to the first page to work out whether $A$ was the category or the object. Of course we can write

   $$\forall \delta \forall x \gt 0 \exists y \gt 0 : \forall \epsilon, |\epsilon - \delta| \lt y \implies |f(\epsilon) - f(\delta)| \lt x$$

   but, frankly, wouldn’t you rather we didn’t?

   John Baez: In lots of ordinary math the strategy of ’bigger fancier fonts for bigger fancier things’ works fine: $a$ for an element of the set $A$ in the category $\mathcal{A}$. But the problem is that in $n$-category theory everything is an object in some $n$-category… including every $n$-category, and every functor between them. So, the idea of a clear hierarchy of two or three types of things, with bigger fancier things getting bigger fancier fonts, breaks down. For example, while you might want to say

   “Consider the functor $\mathfrak{A}: \mathcal{A} \to \mathcal{A}$…”

   you’ll often want to continue that sentence with

   “… and think of it as an object $\mathfrak{A} \in End(\mathcal{A})$. Now, since $End(\mathcal{A}) \in MonCat$, it makes sense to equip $\mathfrak{A}$ with the structure of a monoid object, and we then call it a monad.”

   This is just a very simple example of the kind of fancy level-shifting that takes place. And this level-shifting is precisely the style of thinking that makes $n$-category theory so powerful.

   For this reason, at some point I decided Jim Dolan was very wise to put everything in the same font: it encourages mental flexibility. I later impressed Ross Street greatly when I wrote on the blackboard “Consider a bicategory $x$.”

   Even though I’ll be eternally proud of that moment, I think it can be good to use a couple of different fonts in a given discussion to help ’set the types’ of the entities in question — hence the term ’typesetting’. But this sort of distinction is only helpful locally: it’s not so good to demand globally consistent typesetting. Your big fancy algebraic theory $\mathcal{A}$ is likely to become someone else’s puny little object $a$ in $FinProdCat$, and then $FinProdCat$ will become a puny little object in $Doctrines$, and so on.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 26th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItex(continues) > [[Andrew Stacey]]: I take your point; I was aware that it would have been hard to continue further. Whilst I have an old _Lettraset_ catalogue to draw on for blackboards, sadly it's not implemented in the STIX fonts. But I would like to use the fonts we do have, and so Toby's point is more important that us squabbling over whether or not we should change $\mathcal{V}$ to $V$ or not. > [[John Baez]]: I agree with that. I guess some group of people should argue about whether most of our readers will have various fancy fonts installed... I think we should _not_ assume this. > [[Andrew Stacey]]: If we're going to take this sort of thing into account then we need it clear somewhere since otherwise those of us fortunate enough to have the correct fonts don't know what doesn't display so well on other, more limited, systems! > I did find it getting awkward in writing out the proof that a TW-monoid in abelian groups was a ring to talk of $A$ being an object of $V$. I'd much rather have said $A$ is an object of $\mathcal{A}$. So if we're allowed to use this sort of hierarchy locally, can I change it all back now? > [[John Baez]]: If you insist, but I don't like it: I don't think most reader will find the letter $V$ more 'awkward' than $\mathcal{A}$. And more people will be able to read $V$ than $\mathcal{A}$. > [[Andrew Stacey]]: Linking this to your question above, for something like this where there is a fair amount of notation, what do you think of a "notation used in this page" section for easy reference? > [[John Baez]]: I don't think there's intrinsically more notation in this subject compared to most subjects on the $n$Lab, or more need for fancy fonts. I think that with this subject, as with almost any, it's possible to explain the material very nicely with only a little notation. But that's a kind of pet peeve of mine: I think mathematicians tend to make their work hard to read by introducing more notation than necessary, and not reminding the reader about what it means frequently enough. They like to build towers of thought that are beautiful in principle but not very user-friendly. > [[Andrew Stacey]]: firstly, when I was linking to your question above, I meant to the 'VV^c' question much higher. It was awkward _not_ to use some notation for this category, but it wasn't clear to me that it merited a whole definition to itself. An explanatory sentence is reasonable the first time that it is used, and maybe after a long period with no use, but one of the advantages of hyperlinks is that I could put a "Notation used in this page" section at the bottom, stick in: > * $VV^c$: the category of co-$V$-algebraic objects in $V$ > and put a little hyperlink whenever it is used, something like $VV^c$$([notation](#notation)) (okay, that doesn't work as I haven't created the notation section); possibly with a little semi-quaver instead of the word "notation". > I completely agree with your peeve, but _no_ notation is as horrible as too much notation. Notation should help and guide the reader through a complicated argument, never intruding but always easing the path. > If you take the Dolan principle to its extreme, we should just start the document at the letter $a$ and increase as we proceed. Then the only thing that we have to keep track of is whether or not we have referred to something before or not. Actually, if we combine this with the slogan "Do no evil" then we should _always_ use a new symbol and note where things are isomorphic. Thus > $$ \forall a \forall b \gt c \exists d \gt e: \forall f, |g - h| \lt i \implies |j(k) - l(m)| \lt n $$ where $$ c \cong 0, e \cong 0, g \cong a, h \cong f, i \cong d, k \cong a, l \cong j, m \cong h, n \cong b $$ (I think!) > [[John Baez]]: I never take any principle to extremes, not even this one. I mainly just want to make life easy on people who read my stuff or hear my talks. I think links to "notation used on this page" would be great when used _in addition_ to on-the-spot explanations of the notation when it's introduced, and occasional reminders. I'm afraid however that most mathematicians would use these hyperlinks as an excuse to use more notation, when 99% of them should spend time figuring out how to use less. > Anyway, I think I'm starting to repeat myself, so I'll stop. > [[David Roberts]]: I find myself writing $X$ for a topological groupoid as well as a space, as the latter is one of the former, and I want to replace spaces with groupoids anyway. Ditto with functors (and eventually anafunctors, but that is a little trickier). The move to replace spaces by stacks, which is a good thing, seems to have that psychological barrier of people obeying the urge to call a stack a symbol in an unreadable font. Many-folds were once tricky... > That's my rant over

   (continues)

   Andrew Stacey: I take your point; I was aware that it would have been hard to continue further. Whilst I have an old Lettraset catalogue to draw on for blackboards, sadly it’s not implemented in the STIX fonts. But I would like to use the fonts we do have, and so Toby’s point is more important that us squabbling over whether or not we should change $\mathcal{V}$ to $V$ or not.

   John Baez: I agree with that. I guess some group of people should argue about whether most of our readers will have various fancy fonts installed… I think we should not assume this.

   Andrew Stacey: If we’re going to take this sort of thing into account then we need it clear somewhere since otherwise those of us fortunate enough to have the correct fonts don’t know what doesn’t display so well on other, more limited, systems!

   I did find it getting awkward in writing out the proof that a TW-monoid in abelian groups was a ring to talk of $A$ being an object of $V$. I’d much rather have said $A$ is an object of $\mathcal{A}$. So if we’re allowed to use this sort of hierarchy locally, can I change it all back now?

   John Baez: If you insist, but I don’t like it: I don’t think most reader will find the letter $V$ more ’awkward’ than $\mathcal{A}$. And more people will be able to read $V$ than $\mathcal{A}$.

   Andrew Stacey: Linking this to your question above, for something like this where there is a fair amount of notation, what do you think of a “notation used in this page” section for easy reference?

   John Baez: I don’t think there’s intrinsically more notation in this subject compared to most subjects on the $n$Lab, or more need for fancy fonts. I think that with this subject, as with almost any, it’s possible to explain the material very nicely with only a little notation. But that’s a kind of pet peeve of mine: I think mathematicians tend to make their work hard to read by introducing more notation than necessary, and not reminding the reader about what it means frequently enough. They like to build towers of thought that are beautiful in principle but not very user-friendly.

   Andrew Stacey: firstly, when I was linking to your question above, I meant to the ’VV^c’ question much higher. It was awkward not to use some notation for this category, but it wasn’t clear to me that it merited a whole definition to itself. An explanatory sentence is reasonable the first time that it is used, and maybe after a long period with no use, but one of the advantages of hyperlinks is that I could put a “Notation used in this page” section at the bottom, stick in:

   • $VV^c$: the category of co-$V$-algebraic objects in $V$

   and put a little hyperlink whenever it is used, something like $VV^c$$(notation) (okay, that doesn’t work as I haven’t created the notation section); possibly with a little semi-quaver instead of the word “notation”.

   I completely agree with your peeve, but no notation is as horrible as too much notation. Notation should help and guide the reader through a complicated argument, never intruding but always easing the path.

   If you take the Dolan principle to its extreme, we should just start the document at the letter $a$ and increase as we proceed. Then the only thing that we have to keep track of is whether or not we have referred to something before or not. Actually, if we combine this with the slogan “Do no evil” then we should always use a new symbol and note where things are isomorphic. Thus

   $$\forall a \forall b \gt c \exists d \gt e: \forall f, |g - h| \lt i \implies |j(k) - l(m)| \lt n$$

   where

   $$c \cong 0, e \cong 0, g \cong a, h \cong f, i \cong d, k \cong a, l \cong j, m \cong h, n \cong b$$

   (I think!)

   John Baez: I never take any principle to extremes, not even this one. I mainly just want to make life easy on people who read my stuff or hear my talks. I think links to “notation used on this page” would be great when used in addition to on-the-spot explanations of the notation when it’s introduced, and occasional reminders. I’m afraid however that most mathematicians would use these hyperlinks as an excuse to use more notation, when 99% of them should spend time figuring out how to use less.

   Anyway, I think I’m starting to repeat myself, so I’ll stop.

   David Roberts: I find myself writing $X$ for a topological groupoid as well as a space, as the latter is one of the former, and I want to replace spaces with groupoids anyway. Ditto with functors (and eventually anafunctors, but that is a little trickier). The move to replace spaces by stacks, which is a good thing, seems to have that psychological barrier of people obeying the urge to call a stack a symbol in an unreadable font. Many-folds were once tricky…

   That’s my rant over

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeSep 26th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItex(continues: a pity that a single query box does not fit into the preset limits of a single $n$Forum submission) > _Toby_: Just stepping in to clear the air about what Jim Dolan actually does. Here is an example, not as he would write it, but using symbols as he would write them on a blackboard: > Let $C$ be a category. Let $C_0$ and $C_1$ be objects of $C$. Let $C_2$ be a morphism from $C_0$ to $C_1$. ... > [[Andrew Stacey]]: John, I completely agree with your points but I think that you're being a bit pessimistic. Yes, 99% of mathematicians will misuse a new feature to make their presentations or articles dire. But they would have been dire without those new features. The remaining 1% should be encouraged to use the new features to make their already-excellent stuff even better. The great thing about the wiki-format is that _someone else_ can go along later putting in those notational footnotes. Indeed, it is _better_ if it is someone else because it is clearer to the non-author where they should be. After all, I know what $VV^c$ means when I write it so I don't need reminding. You, or someone else, aren't so familiar with the notation so can - and should - put in the little reminders. What makes this easiest is if I have _already_ put a little notational summary at the bottom of the page so that all you have to do is add the correct link wherever you think it necessary. > Of course none of this excuses me explaining the terms correctly and thinking carefully about what notation I ought to use. But just because it has the potential to be horribly misused doesn't mean that we should shun it if it also has the potential to make stuff much better. > The fact that this stuff is on the web should make a difference. Otherwise we may as well just have a load of PDFs of LaTeX documents. The big addition of the web is the ability to link documents and to navigate back and forth between those links. We should be experimenting with this and trying out different ways to implement it. > Which has gotten us a long way from the argument as to whether or not $\mathcal{V}$ is a category or not! > My principle, which I'm quite happy to take to any extreme, is: whatever aids comprehension is Good, whatever hinders it is Bad. Careful choice of notation is Good, "fancy fonts for fancy gizmos" is _by itself_ Bad. > On the issue as to whether or not we should use $\mathcal{A}$ and so forth, I'm less inclined to be kind to our Dear Readers. We already insist that they be able to read MathML. That puts a fairly high barrier on entry, I don't think that requiring the fonts puts that much higher. If you do want to impose this rule, then someone needs to do some extensive testing to find out what is and isn't allowed.

   (continues: a pity that a single query box does not fit into the preset limits of a single $n$Forum submission)

   Toby: Just stepping in to clear the air about what Jim Dolan actually does. Here is an example, not as he would write it, but using symbols as he would write them on a blackboard: Let $C$ be a category. Let $C_0$ and $C_1$ be objects of $C$. Let $C_2$ be a morphism from $C_0$ to $C_1$. …

   Andrew Stacey: John, I completely agree with your points but I think that you’re being a bit pessimistic. Yes, 99% of mathematicians will misuse a new feature to make their presentations or articles dire. But they would have been dire without those new features. The remaining 1% should be encouraged to use the new features to make their already-excellent stuff even better. The great thing about the wiki-format is that someone else can go along later putting in those notational footnotes. Indeed, it is better if it is someone else because it is clearer to the non-author where they should be. After all, I know what $VV^c$ means when I write it so I don’t need reminding. You, or someone else, aren’t so familiar with the notation so can - and should - put in the little reminders. What makes this easiest is if I have already put a little notational summary at the bottom of the page so that all you have to do is add the correct link wherever you think it necessary.

   Of course none of this excuses me explaining the terms correctly and thinking carefully about what notation I ought to use. But just because it has the potential to be horribly misused doesn’t mean that we should shun it if it also has the potential to make stuff much better.

   The fact that this stuff is on the web should make a difference. Otherwise we may as well just have a load of PDFs of LaTeX documents. The big addition of the web is the ability to link documents and to navigate back and forth between those links. We should be experimenting with this and trying out different ways to implement it.

   Which has gotten us a long way from the argument as to whether or not $\mathcal{V}$ is a category or not!

   My principle, which I’m quite happy to take to any extreme, is: whatever aids comprehension is Good, whatever hinders it is Bad. Careful choice of notation is Good, “fancy fonts for fancy gizmos” is by itself Bad.

   On the issue as to whether or not we should use $\mathcal{A}$ and so forth, I’m less inclined to be kind to our Dear Readers. We already insist that they be able to read MathML. That puts a fairly high barrier on entry, I don’t think that requiring the fonts puts that much higher. If you do want to impose this rule, then someone needs to do some extensive testing to find out what is and isn’t allowed.

5. • CommentRowNumber5.
   • CommentAuthorAndrew Stacey
   • CommentTimeSep 27th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexI've removed the other query box (I don't think it's worth archiving other than in the page history).

   I’ve removed the other query box (I don’t think it’s worth archiving other than in the page history).