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• CommentRowNumber1.
• CommentAuthorPeter Heinig
• CommentTimeJul 6th 2017
• (edited Jul 8th 2017)

It would not surprise me if there already is a thread for this, but none was found: this thread is meant to collect (useful!, serious!, sparingly used!, carefully chosen!) verbal mnemonics/slogans/unusual-yet-useful-technical-terms relevant to category theory (especially higher-category theory). Some of them can also be a bit cryptic, and given without much explanations, such as the example below.

The emphasis should be more on compressed mnemonics than on sweeping slogans (nice though those can be).

This is not to get into a discussion about whether mnemonics are good or bad or tasteless.

I, too, am of the opinion that, by and large. structural thinking and “abstrakte Anschauung” is to be preferred to grade-school-like verbal crutches. In Dutch and the German the vernacular term for “mnemonic” is even inherently disparaging, by the way. And then again, mnemonics is sort of an art, with a venerable tradition since antiquity.

Here is an example, perhaps not even a particularly nifty one, to get this thread started:

(monoidal monoidoid)$=$(monobject bimonoidoid)

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJul 6th 2017

ugh.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJul 6th 2017

I think Lawvere and others used to be fond of “slogans” (on the order of “Adjoint functors arise everywhere”), but my vague impression is that was mostly back in the 60’s, man, when there was a revolutionary excitement in the air. These were once discussed at the Café, here.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeJul 6th 2017

I always found RAPL(=right adjoints preserve limits) useful

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJul 6th 2017

I think of “right adjoints preserve limits” as more of a theorem than a slogan.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeJul 7th 2017

@mike I think of the acronym as a mnemonic :-)

• CommentRowNumber7.
• CommentAuthorPeter Heinig
• CommentTimeJul 8th 2017
• (edited Jul 8th 2017)

Perhaps I should have left out “slogan” out of the title (and now retroactively did).

The idea is to have this collection rather focused on compressed mnemonics than on sweeping slogans.

I expect to the number of “entries” here to be rather small, if one sticks to including sweeping slogans sparingly, if at all. Of course, like already mentioned, it is delicate matter whether to share mnemonics, and more often than not, they should probably be remain a private matter.

The example given by me above is self-explanatory, and uses a widely known coinage which seems to a decade old at least. Of course one can grind out more of that kind, verbalizing the “periodic table”, but there seems to be little point is doing so; the one given (to me) is sufficient to remind me of (0) oidification, (1) the periodic table, (2) Cat-likeness versus Mod-likeness.

RAPL is an appropriate example. It is not very compressed or fecund (it seems one can manipulate it in fewer ways than monoidalmonoidoid=monobjectbimonoidoid) or unusual, but certainly useful sometimes.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 8th 2017
• (edited Jul 8th 2017)

One basic point about category theory which might usefully be highlighted more to newcomers and not-so-new-comers is that the original sequence of motivations (here):

• the point of categories is to be able to speak of functors;

• the point of functors is to be able to speak of natural transformation

should really have one more step:

• the point of natural transformations is to speak of adjunctions.

It is really with the concept of adjunction that category theory becomes a theory, and where its beauty and richness resides.

(Keeping in mind here that universal constructions such as limits and colimits are all examples of adjunctions, as is the concept of equivalence of categories, etc. )

Adjunctions, in turn, may arguably been regarded as the formalization of the concept of duality (Lambek 81).

So: category theory is duality theory.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeJul 8th 2017

Keeping in Mac Lane’s slogan “All concepts are Kan extensions”, and the fact Kan extensions can be reduced to adjunctions, Urs is pretty much right on the money.

• CommentRowNumber10.
• CommentAuthorPeter Heinig
• CommentTimeJul 13th 2017
• (edited Jul 13th 2017)

A very basic one (usable e.g. in a first lecture in a first course on category theory—not saying that this were little known or of use to anyone round here personally, just mentioning it since it seems to fit the remit of this thread):

($\Rightarrow) =$(natural tranformation)

There is a Leibnizian felicity in this notation, like $\frac{dy}{dx}$ it conveys something essential about the idea: here, that a natural tranformation is some-sort-of a family of arrows while not being the same as this, of course, like a derivative being some-sort-of fraction, without being the same as this. If only the notation $\Rightarrow$ would not clash with the traditional notation for implication, which most audiences have encountered earlier, and if only this notation would “scale” better (in more than one sense of the word).

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeJul 13th 2017

That’s not how I think of it. I think of it rather in terms of dimensionality. For the 2-category $Cat$, we use $\to$ for 1-cells = functors and $\Rightarrow$ for 2-cells = natural transformations. It would be nice if we could do 3-cells in the same matter (or maybe we can and I don’t know it).

• CommentRowNumber12.
• CommentAuthorRodMcGuire
• CommentTimeJul 14th 2017
• (edited Jul 14th 2017)

It would be nice if we could do 3-cells in the same matter

• $\to$ - 1-cells
• $\Rightarrow$ - 2-cells
• $⇛$ - 3 cells

but to get the triple arrow you have to ask for it as unicode - it doesn’t have an iTeX name and only comes in left/right orientations ($⇛$, $⇚$) while the lesser arrows have vertical and diagonal shapes - $⇑$, $⇓$, $⇖$, $⇗$, $⇘$, $⇙$.

arrows block

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeJul 14th 2017

I’ve seen people use triple arrows for 3-cells sometimes in diagrams in papers; probably you can make it with tikz. I’ve also seen people write a “3” on top of the arrow.

• CommentRowNumber14.
• CommentAuthorPeter Heinig
• CommentTimeJul 14th 2017
• (edited Jul 20th 2017)

While this thread is still on the topic (one never knows where threads will spin to), and give an application of the notation that Mike mentioned, i.e., numbers written near to arrows as a method to symbolize higher cells: [removed; too playful]

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeJul 14th 2017

Peter, if you click on Source, you can copy over their responses to the new thread yourself.

• CommentRowNumber16.
• CommentAuthorPeter Heinig
• CommentTimeJul 14th 2017
• (edited Jul 15th 2017)

Re #14:

• The code is what some would call “hackish”. The horizontal alignment is suboptimal. Both are concessions to the constraints of the software of this forum.

• The question marks are to say that both the “dimensionality” and the “invertibibility” of the higher cell are an object of current research.

• CommentRowNumber17.
• CommentAuthorPeter Heinig
• CommentTimeJul 18th 2017
• (edited Jul 18th 2017)

(category of adjunctions)$\hookrightarrow$(double categories)

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeJul 18th 2017

I expect to the number of “entries” here to be rather small, if one sticks to including sweeping slogans sparingly, if at all. Of course, like already mentioned, it is delicate matter whether to share mnemonics, and more often than not, they should probably be remain a private matter.

Yes. The potential for self-indulgence in this thread seems evident. Personally I feel it should be brought to a close.

• CommentRowNumber19.
• CommentAuthorPeter Heinig
• CommentTimeJul 18th 2017
• (edited Jul 18th 2017)

The potential for self-indulgence in this thread seems evident. Personally I feel it should be brought to a close.

Certainly with a comment like this in it. It hurts (me) to read it.

But on the whole I tend to agree, especially if you mean a generalized potential of this thread to veer in a bad direction, and I share your concern.

Actually, my main concern is not so much self-indulgence (whatever this means in this context), but rather that this thread encourages not engaging in cogent, rational discussion, using more or less standard English, possibly with mathematics added. This is my main concern.

The thread could degenerate into steganographic symbol-flashing, not beholden to the etiquette of using rational English. This was totally unforeseen when I started the thread. I was hoping it could collect lots of useful, precise and compressed mnemonics.

So again, I share your concern (though not the perhaps implicit accusation), and would rather prefer to see the thread removed, not closed.

[Technical note on the mnemonic (category of adjunctions)$\hookrightarrow$(double categories): arguably, Adj this is the most memorable purely category theoretic example of a double category, in particular since it involves the central concept of adjunctions; the category Mod is of course a standard example too, but is of a more algebraic character, and in other basic purely category-theoretic examples of double categories, the vertical arrows (to me) are less distinctive.

And of course, as a concession to the brevity a mnemonic should have, saying “adjunctions” in “category of adjunctions” is not quite right, at least if one takes “adjunction” to mean “fully specified adjunction”, i.e., pair of adjoint functors together with a pair or specified unit and counit. The category Adj, like I learned it, does not expressly involve unit and counit.)

• CommentRowNumber20.
• CommentAuthorTodd_Trimble
• CommentTimeJul 18th 2017

We have a way of “removing” threads, called The Cave, where threads are not erased but they are removed from public sight. This has been used in the past when things have gotten ugly. In the present case it will likely appear to outside eyes that any “ugliness” is coming from my end.

Way in the past there was more encouragement of free-wheeling discussion at the nLab and the nForum. But as the nLab has grown and times have changed, we’ve had to tighten up a bit. Too many people have tried to use the nLab or nForum in ways which drain time and attention of others away, with unfortunate consequences. But your last comment indicates that you fully understand that point and the need for discipline in discussions, which is encouraging – thanks for that.

I am sorry for the hurt my blunt words have occasioned.

In case anyone else wants to add some concluding thoughts, I suggest keeping this thread open just a bit longer, before making arrangements to consign to the Cave.

• CommentRowNumber21.
• CommentAuthorPeter Heinig
• CommentTimeJul 20th 2017
• (edited Jul 20th 2017)

Re #20:

But your last comment indicates that you fully understand that point and the need for discipline in discussions, which is encouraging – thanks for that.

Yes, and I would prefer to not have it moved to “The Cave” but either remove it or, probably better, just leave it at that (with the expectation that it be not continued).