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Given that the axioms of differential cohesion bear so much fruit now, I thought I should try to organize and display them in a clear way, such as to make them visible at one glance and clarify their interaction with those of bare cohesion. For, that interaction is somewhat subtle: the axiom sets look alike, except for a subtle off-set, of sorts, which makes all the difference.
When I attempt to bring that out on paper, I find myself ending up drawing symbols that are reminiscent of the I Ching. Have a look to see what I mean, it’s kind of curious:
How to read those diagram:
the small dot is the base context;
the fat dot is the cohesive context;
the star is the differential cohesive context;
the base-line is a direct image $\infty$-functor, going left to right;
a line above is left adjoint, a line below is right adjoint;
in particular adjacent lines are $\infty$-functors of opposite direction;
a broken line is a fully faithful $\infty$-functor.
What does looking at those diagrams make you think? Suggestions for further/better organization are welcome.
Would look better with arrowheads.
Hi David,
thanks, for the feedback. But I am not sure what to make of your remark. Do you mean to comment on typography, or is there a hidden technical suggestion?
Let me maybe expand a bit on what I am after, to clarify myself.
Part of the point here is that it is noteworthy that the axioms of differential cohesion involve precisely 6 “higher modalities” that come in two adjoint triples
$(Red \vdash \mathbf{\Pi}_{inf} \vdash \mathbf{\flat}_{inf})$
$(\mathbf{\Pi} \vdash \flat \vdash \sharp)$.
where $\mathbf{\Pi}$, $\mathbf{\Pi}_{inf}$ and $\sharp$ are monadic, while $Red$, $\flat_{inf}$ and $\flat$ are comonadic. All are idempotent. And now there is some interrelation between these.
What’s a good succinct way of stating this data? Is this just a bunch of random data, or is there some pattern? Is there maybe a nice natural “Yoga of Six Higher Modalities” underlying this? What’s the slickest way to tell an outsider what this structure is? What’s going on?
That’s the kind of questions I am after. When playing around with this, I found it noteworthy that when writing out the corresponding adjunctions of these (co)monads, they do display something that certainly looks like a pattern. In the entries on differential cohesion these are displayed all with arrowheads and everything. But the direction of the arrowheads is implied already by the position of an arrow in the array. So for pattern-recognition, I found it useful to precisely discard this kind of redundant information.
Doing so shows, as that little pdf note is supposed to do, that some curious binary pattern is at work. One way to say it is that differential cohesion is the system of higher modalities encoded by the binary pattern
$\array{ - & + \\ + & - \\ - & + \\ + & - } \,.$Is that not curious? Maybe I am hallucinating, but I thought that’s something to ponder.
So, do you have answers to these questions:
What’s a good succinct way of stating this data? Is this just a bunch of random data, or is there some pattern? Is there maybe a nice natural “Yoga of Six Higher Modalities” underlying this? What’s the slickest way to tell an outsider what this structure is? What’s going on?
Might we come to see the modality pattern as fundamental, and its realization in terms of smoothness as secondary?
Might we come to see the modality pattern as fundamental, and its realization in terms of smoothness as secondary?
That’s what I am after, all along. To formulate as much as differential geometry/differential cohomolgy axiomatically as possible.
I enjoy this kind of change of perspective. Can we make any progress here? What kind of considerations can play the role of motivating an axiomatisation in terms of this modality pattern? Will it be of the hypothetico-deductive kind - this is a simple axiomatisation which has as consequence something we have found we need in our theories - or could there be some kind of non-empirical a priori justification?
What kind of considerations can play the role of motivating an axiomatisation in terms of this modality pattern?
So we are trying to understand higher structures in higher geometry. For instance the question:
“Since the 1-geometric quantization of the Chern-Simons functional yields 3d Chern-Simons QFT as an ordinary (unextended) QFT, what would be a 3-geometric quantization of the functional that yields the 3d extended QFT as a 3-functor?”
In all these kinds of questions we understand something in low homotopical/categorical degree and are trying to find the sensible generalization to higher homotopical/categorical degree in the hope that this will make us better understand the traditional setup.
And so as a bridge we need an axiomatization that
a) makes natural sense fully generally and
b) reduces to the known structure in the suitable special case.
That’s the kind of game that we have been playing all along. The emphasis on the smooth model is due to point b). It’s a continuous back and forth between these two points and feedback of the first results obtained in higher degree:
we guess a general abstract axiom, check that it reduces to the right low-degree structures, work out what it does more generally. If that fits the bill, we gain trust in that axiom and will use it for formulating other axioms. If things don’t come out right, we go back and fiddle with the axioms.
We feel that things ar going in the right direction if we need very few and very natural fundamental axioms to get out a whole lot of traditional and not so traditional structure. So if we now see that at the bottom of it the axioms fit into a nice pattern as indicated above while giving rise to loads of good theory, that is reassuring and makes us want to record the outcome so far in a nice fashion.
Back in the old days, when I didn’t know better, I used to call that yoga “arrow theory”, since the axioms are formulated in terms of certain universal constructions incarnated as certain diagrams. But nobody else picked up on such “arrow theory” for differential geometry and physics. Then at some point it turned out that what I called “arrow theory” is essentially what other people rather call “categorical semantics of type theory”, and ever since then the communication is much better :-)
Despite hearing the phrase ’[Grothendieck’s] six operations’, I haven’t seen a good explanation that I can link to now, but it seems like something worth comparing this idea to. Especially given recent work that shows that we can embed schemes (or what have you) into higher categories Tannakian-style, I wonder whether there is some sort of relative cohesion going on there. Note that this is just thinking out loud, and could fall down at the first step…
But the direction of the arrowheads is implied already by the position of an arrow in the array. So for pattern-recognition, I found it useful to precisely discard this kind of redundant information
Well, the information may not be strictly necessary to display, but it helps for comprehension. I reminded of Edward Tufte’s redesigns of the classic box plot (see here for a comparison: the traditional one is on the left, Tufte’s versions the middle and right). They contain all the information of the original, but at least for me the meaning does not pop out at a glance, even though I know how to read them. Given that they haven’t really taken off in fields where they could be used, I guess that the same is true for other people.
It is better design to be able to be able to tell the direction of an arrow at a glance, rather than having to count a bunch of parallel lines, especially if they have distinguishing features (the gaps) which are inconsistent between diagrams.
Despite hearing the phrase ’[Grothendieck’s] six operations’, I haven’t seen a good explanation that I can link to now,
Try the $n$Lab: six operations. :-)
So Grothendieck’s six operations are a geometric morphism that extends to an adjoint quadruple (so far this is like cohesion), plus two more: the hom and tensor functor (and there the analogy breaks down) and without locality and connectedness assumptions (and there it goes out of the window).
So this is not the same. But of course it has a certain similar smell to it, which is why I alluded to it by making up, half-jokingly, the term “yoga of six higher modalities”.
Oh, and in the file with the diagrams, they don’t seem to line up, which means I can’t even consistently figure out which diagram is supposed to correspond to which label. So while this at present is a handy mnemonic for the expert(s), it doesn’t help others (i.e. me) very much.
A comment on spelling: it took a while before I realised i ging referred to I Ching (or Yi Jing, if one is to concede to the pinyin-ists)…
@David:
they don’t seem to line up
so by the rules of #1 the base line is always the direct image functor. For instance local connectedness is given by an extra left adjoint, and so there is a triple of lines with the base line being the lowest. For locality there is an extra right adjoint, and so in this case the base line is the middle line.
This is getting somewhat interesting as we come to differential cohesion. In the first version we think of the quadruple of lines as an essential embedding of the cohesive context $\bullet$ into its infinitesimal neighbourhood $\star$ and so the base line is the second line from below. But crucially we may “turn around and shift by one” and regard the same diagram, but shifted, as an “exceptionally essential” morphism from $\star$ to $\bullet$ (this is the very last diagram in the pdf), which is the way to see that differential cohesion equips $\star$ with a “contraction in two steps” (case eight in the pdf). The first of these, the relative contraction, interprets as the infinitesimal path $\infty$-groupoid, the second as the remaining finite path $\infty$-groupoid.
@Zhen Lin
it took a while before I realised i ging referred to I Ching
Oh dear, stupid me. Thanks for correcting this! (I followed the German transcription and then ignored the capitalization.)
I have fixed it now in the above comments.
@David, re #8,
I only now see that you added something to your comment. So I’ll reply now to the piece that I hadn’t seen before:
the arrowheads and everything have been displayed all along, at cohesion and infinitesimal cohesion. This here is not about suggesting that we never draw them again. This here is about saying: look, if we change perspective for a second and redraw all those diagrams that we have been playing with all along by simply only remembering which moprhism is fully faithful and which is not, then a curious pattern is brought out, which isn’t so vividly visible in the standard denotation.
Concerning your remark about “inconsistency”: can you say what you find inconsistent?
Just looking at it now, I realise that the diagram corresponding to the label ’discrete’ consists of two I Ching-style diagrams, and ditto for ’differentially cohesive’. I just thought that you’d done a really sloppy (late night) latexing job so that nothing quite lined up (apologies!)
’Inconsistent’ just meant that I wasn’t sure of how the alignment of text and diagram was supposed to work, not the actual conceptual/mathematical content being inconsistent.
It makes a lot more sense now! It would be good if the line which was at the level of the markers $\bullet$, $\ast$ and $\cdot$ was thicker, to provide a visual confirmation of the vertical alignment of the markers. Also, perhaps replace $\cdot$ with $\circ$ (I know it’s against the general philosophy of the topos as a fat point…)
Okay, thanks David!
I thought about these typesetting questions a bit. The only reason that “$\bullet$” etc appears is to indicate the base line. I thought about making it fat, but somehow that destroyed the (what I found) nice look.
I think what I’ll do is draw long faint horizontal pointed lines all across the page, as in a notebook, and then arrange the symbols on them.
But this has to wait until a little later today. I still need to urgently fill some remaining gaps in category object in an (infinity,1)-category for my seminar talk today… :-)
Now I had a spare minute to expand that pdf, adding a bit more explanation of what is meant to be going on and (hopefully) improving the notation. It’s here:
(I would do all this in an nLab-entry on my personal web, but I’d need to think about how to the diagrams there…)
If you’re using $\Box$ to remind us of the ’necessarily’ modality, shouldn’t we have $\diamond$ for ’possibly’ too? The former for comonadic, the latter for monadic.
Good point, thanks. I have changed it accordingly.
That looks much nicer! The solution of having a faint baseline is better than my suggestion. It’s perfectly clear now.
Are you going to start pushing the I Ching similarity further?
The solid line represents yang, the creative principle. The open line represents yin, the receptive principle.
Fully faithful functors are receptive, while ordinary functors are creative?
But, seriously, what do you see in the diagrams? Could it be possible to raise the baseline further so it passes through the top lines?
seriously, what do you see in the diagrams?
it’s something very mundane combined with a bit of fun. The mundane part is: I found myself repeatedly getting myself mixed up about which monad goes where and factors through which other monad in a context of differential cohesion, with its eight adjoints. So one day I sat down with the intent to write it out clearly once and for all, so that I would never get myself mixed up again. In the course of this I noticed that I Ching-like pattern (obvious as it may have been all along). And that helped a lot. Now indeed it’s a breeze for me to remember everything and I’ll ever get my infinitesimal paths mixed up again.
The fun part is then hard to resist: play further with that I Ching-analogy, maybe over a beer after a long day. Clearly one shouldn’t push it too far. But one striking similarity just as clearly remains, for whatever it’s worth: it’s just a fact that simply by placing those lines in different positions, we encode, via higher topos theory, a multitude of “flavours of geometry” as indicated in the above file.
If nothing else, that makes for a pretty “illustrated table of contents” for a document on cohesion…
If nothing else, that makes for a pretty “illustrated table of contents” for a document on cohesion…
This is now in differential cohomology in a cohesive topos (schreiber), currently p. 152.
it’s just a fact that simply by placing those lines in different positions, we encode, via higher topos theory, a multitude of “flavours of geometry” as indicated in the above file.
So does the set of flavours look complete to you? I mean it’s not like you’re in the situation where you just set up the periodic table and noticed some elements missing and so go looking for them? Is it possible to give reasons why the table could not be extended?
Hi Urs,
your hexagrams look great (I should read the nforum more frequently)! When I got confused over composition and direction of the adjoints tuples I just piled the tuples, e.g. $\array{(i_!\dashv i^*\dashv i_* \dashv i^!)\\(j_!\dashv j^*\dashv j_* \dashv j^!)}$. But your notation is more expressive and discards irrelevant typographical elements - and it is also much more cultic.
Is it possible to give reasons why the table could not be extended?
Did someone indicate that it could not?
more expressive …. much more cultic.
:-)
Is it possible to give reasons why the table could not be extended?
Did someone indicate that it could not?
Exactly. So far the table ends where I am in loss of further words to assign to diagrams. But maybe we find more useful identifications.
where I am in loss of further words to assign to diagrams
I had not in mind something so “hands-on”. If we have an “∞-gram”
$i_1\dashv i^1\dashv i_2\dashv i^2\dashv\dots$instead of an hexagram of adjoints, there are lots of derived monads and comonads and compositions thereof.
If we assume that the $i_j$ are full and faithful we also have
$i_1\stackrel{\eta_12 i_1}{\to} i_2 i^1 i_1\stackrel{\sim}{\to} i_2\stackrel{\eta_23 i_2}{\to}i_3 i^2 i_2\stackrel{\sim}{\to}i_3$Natural questions would then be if this game becomes trivial at some point and when functors “in the difference” of the categories between which this “n-grams” go are compositions of some of the occurring adjoints or monads.
I wasn’t suggesting one way or the other whether the table is complete. It’s much more interesting to think it isn’t, then like Mendeleev we can go looking for eka-silicon and dvi-manganese.
So which diagrams do you have that you can’t find words for?
So which diagrams do you have that you can’t find words for?
In a way there are actually not so many, really. Because the way to get more diagrams is to add more lines. But every further line added is a very strong further condition and brings the system, roughly, “close to being trivial” (the discrete case). (Maybe it would be good to try to formalize this statemen.)
One term that I could add is maybe for four non-broken lines. In view of the non-broken three-line diagrams, this might be called “locally cohesive”. But not sure, maybe that needs a different word.
What is probably more interesting than adding more lines is making more compounds, i.e. putting the diagrams next to each other, as for infinitesimal cohesion. I think a sequence of length-$n$ of the four-line diagrams will give infinitesimal cohesion with $n$ orders of infinitesimals.
But otherwise, maybe I want to take back where I suggested above that we can greatly extent the table. There is probably not that much room, actually, in that the further possibilities would “tend to be trivial”.
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