A puzzle:
Name as many instances as you can of mathematicians or physicists who introduced a logical sequence of three concepts, each depending on the previous one, where the third was crucial for the applications they had in mind.
Posted at January 29, 2009 6:14 AM UTCTrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1902
Eilenberg and Mac Lane introduced the notion of natural transformation in their 1945 category theory paper with applications to algebraic topology in mind.
To define natural transformations they introduced functors.
To define functors they introduced categories.
Yes, that’s one of the examples I had in mind! Great!
But there’s another, even more familiar. I only noticed the resemblance recently.
Albert Einstein.
(1) Special Relativity;
(2) General Relativity;
(3) The Unified Field Theory of 1957 which, after Godel’s time travel solutions to the field equations, was used to extend his life and win his second Nobel Prize.
Whoops, (3) is that not on the branch of the multiverse where we reside…
It was a community, not a single person, who established this sequence of three formulations of classical mechanics:
(1) Newtonian,
(2) Lagrangian, and
(3) Hamiltonian.
For a long time I’ve enjoyed this example:
Saunders Mac Lane is said to have remarked, “I didn’t invent categories to study functors; I invented them to study natural transformations.” The reason is that mathematicians were comparing different cohomology theories, and needed a precise concept of when two such theories were ‘naturally isomorphic’.
One reason this example is noteworthy is that it means Eilenberg and Mac Lane were really inventing the 2-category of categories. This illustrates the ‘upwards pull’ of categorification: to really understand -categories, you need the -category of all -categories, which means you need to understand -categories.
But yesterday, while teaching an undergraduate class, I noted another trio of concepts in which the ‘highest’ and least obvious of the three is really the one needed in the most glorious application of the theory.
What is it?
I like ‘special relativity, general relativity, unified field theory’ — but that’s not it, and unfortunately Einstein never really found the third item.
Hmm, you’re teaching differential geometry aren’t you? A rich area for towers of concepts.
I remember being shocked at the definition of a tangent vector as an equivalence class of paths.
Is that the particular example you are after? Or is it curvature or geodesic?
You need the three concepts to be devised by the same person? Gauss or Riemann then.
I was thinking of something along the lines of Jonathan’s suggestion except I would go
- Newtonian
- Hamiltonian (symplectic)
- 2-symplectic
Or something…
You guys are going to be disgusted when I finally reveal my answer. You’re thinking about stuff that’s way too fancy. I’m thinking about a trio that rolls off the tongue as easily as “category, functor, natural transformation” — but one that many more people know about.
Newton again, finally with application to rocketry:
I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.
II. The relationship between an object’s mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors (as indicated by their symbols being displayed in slant bold font); in this law the direction of the force vector is the same as the direction of the acceleration vector.
III. For every action there is an equal and opposite reaction.
That’s interesting, but Newton’s three laws are not “a logical sequence of three concepts, each depending on the previous one, where the third was crucial for the applications he had in mind.”
Neither are the three laws of thermodynamics.
Give us a hint — what field?
You know the answer to this question, so you don’t need a hint.
Again, creation by a community:
Krippendorff, Klaus. “Information” Paper presented at the annual meeting of the International Communication Association, TBA, San Francisco, CA, . 2008-12-11
Abstract:
Much contemporary communication research stands on the shoulders of three mathematical theories: the logical, statistical, and algorithmic theories of information. Logical theory concerns distinctions and selections among logical possibilities. As such, it addresses a semantic space of possible realities. Statistical theory concerns observed probabilities, and is familiar from Claude Shannon’s theory of communication. Algorithmic theory concerns the information that is required to compute a function or problem (i.e., some software requires more memory than others). Each of these theories center on the distinctions (possibilities, probabilities, and length of instructions) which, from the human perspective, give rise to meaning. The translation of distinguishable signals into informative messages depends on at least three aspects of their context: (i) temporality (e.g., a comprehensible sequence); (ii) rules (e.g., situational constraints); (iii) a history of relationships between senders and receivers; and (iv) the known intentions of senders and needs of receivers.
I think Krippendorf is using ‘at least three’ to mean ‘four or more’. I want a sequence of three concepts, in clear logical succession.
As the Second Brother put it:
Three shalt be the number thou shalt count, and the number of the counting shall be three. Four shalt thou not count, nor either count thou two, excepting that thou then proceed to three. Five is right out.
From mathematical finance:
1.) Black-Scholes (Stock options from “point” prices)
2.) LIBOR Market Model (Bond options from “yield curve” prices)
3.) Not established yet, but I would like to :)
Since I mentioned Kurt Gödel [28 April 1906 - 14 Jan 1978]:
(1) doctoral dissertation under Hahn’s supervision in 1929 submitting a thesis proving the completeness of the first order functional calculus;
(2) Gödel’s Incompleteness Theorems”. In 1931 he published these results in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved;
(3) His masterpiece Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940) is a classic of modern mathematics. In this he proved that if an axiomatic system of set theory of the type proposed by Russell and Whitehead in Principia Mathematica is consistent, then it will remain so when the axiom of choice and the generalized continuum-hypothesis are added to the system. This did not prove that these axioms were independent of the other axioms of set theory, but when this was finally established by Cohen in 1963 he built on these ideas of Gödel.
Reference for above is Kurt Gödel, by J J O’Connor and E F Robertson
1) Heavens, 2) Earth, 3) Light.
Now that’s a good answer! 
In the beginning God created the heaven and the earth.
And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
And God said, Let there be light: and there was light.
However, it’s not the one I had in mind. I was definitely thinking about human-made mathematics or physics.
Given that John is teaching Differential Geometry:
(1) Vector bundle
(2) Vector bundle with connection
(3) The Riemannian connection on the tangent bundle
I know that I found Diff. Geom. got much easier when I started classifying results according to which of these three concepts they were really about. It doesn’t exactly roll off the tongue, though.
Sorry, I don’t think first anybody invented vector bundles, and then vector bundles with connection, and then the Riemannian connection on the tangent bundle… with his goal being to study the third one. These ideas may seem like a logical sequence now, but in fact the last of them was probably invented first.
Also, I don’t see them as a natural progression with same sort of inevitability as ‘category, functor, natural transformation’.
Hmm…
pointed homotopy of pointed paths, homotopy *class* of a pointed path, universal covering space?
Okay, since I mentioned Hamilton:
(1) Reals;
(2) Complex;
(3) Quaternion
(4) bonus, by Cayley et al.: Octonions
“On a new Species of Imaginary Quantities connected with a theory of Quaternions”
By Sir William R. Hamilton
[Proceedings of the Royal Irish Academy, Nov. 13, 1843, vol. 2, 424-434]
It is known to all students of algebra that an imaginary equation of the form i^2 = -1 has been employed so as to conduct to very varied and important results. Sir Wm. Hamilton proposes to consider some of the consequences which result from the following system of imaginary equations, or equations between a system of three different imaginary quantities:
i^2 = j^2 = k^2 = -1.
ij = k, jk = i, ki = j.
ji = -k, kj = -i, ik = -j.
no linear relation between i, j, k being supposed to exist…
Sir W. Hamilton calls an expression of the form Q a quaternion; and the four real quantities w, x, y, z he calls the constituents thereof. Quaternions are added or subtracted by adding or subtracting their constituents….
Leray invented (1) spectral sequences to compute (2) sheaf cohomology and for that he had to invent (3) sheafs.
Another possibility is
1)Manifolds
2)Differential forms
3)Stokes theorem
The first two don’t really go in order here, but maybe
I feel a strange blend of amusement and despair.
When you see the trio I’m thinking of, you’ll recognize it as forming an inevitable logical sequence, as fundamental as “1,2,3” or “category, functor, natural transformation” — and just as important in mathematics and physics. And as with “category, functor, natural transformation”, the third member of this sequence is its crowning glory.
i)Vectors
ii)Vector spaces
iii)Linear morphisms
Perhaps a slightly better variant on this might be (I don’t actually know the history well enough):
1. Limits
2. Differentiation
3. Integration
Certainly most of the big applications of calculus use integration, but you can’t compute them without derivatives, and those require limits. (Though I’m not sure if Newton thought about it that way.)
A dark horse candidate, and almost certainly not what you had in mind… but it does roll of the tongue.
Claude Shannon
1) Information
2) Compression
3) Capacity
When someone finally guesses the trio I have in mind, everyone else will probably get angry. They’ll probably argue that the first — and maybe even the second — concept in this trio was not really invented by the person who put the third to such astoundingly effective use.
And in a sense they’ll be right. But surely, the whole formalism that makes this trio into a tightly connected sequence was first clearly understood by the person in question. And surely everyone here is familiar with what I’m referring to. So I don’t feel at all guilty for posing this puzzle.
Integers, sets, groups; Galois
Nice, but much more erudite than the one I have in mind… and I don’t see these three concepts as forming an inevitable progression, where each one leads to the next in a systematic way.
You’re going to be really annoyed when someone comes up with the answer I’m thinking of.
Is it: 1) Tangent, 2) Normal, and 3) Frenet frame?
I feel sure this isn’t right, but I’ll throw it out there:
All three are important in both math and physics, with the third the most important, the crowning glory.
Position, velocity, acceleration; Newton?
Dangit, I just had that exact same thought.
I know these are not what you have in mind. I am not even sure of the accuracy as I am not sure what name to put to them. How about?
1) vector space
2) dual vector space
3) cohomology
or
1) Hilbert space
2) self-adjoint operator
3) eigenstate
You’re not thinking of something as simple as:
addition, multiplication, exponentiation
are you?
(Even if you’re not, this suggestion might get people to aim a bit lower on the sophistication scale!)
After posing the puzzle I did indeed think about addition, multiplication and exponentiation as a famous trio of concepts in logical progression. It’s very nice, and you’re right — it’s closer to the level of sophistication I was looking for. But it doesn’t quite match my puzzle: “Name as many instances as you can of mathematicians or physicists who introduced a logical sequence of three concepts, each depending on the previous one, where the third was crucial for the applications they had in mind.”
Several examples come to mind:
(1) Integers, rational, reals.
(2) Chart, atlas, manifold
(3) Subset, open set, continuous map.
Based on David’s detective work, I think JB had (2) in mind.
Todd wrote:
Position, velocity, acceleration; Newton?
YES!!!
Each one is clearly obtained from the previous one in a systematic way, and Newton invented this way — differentiation!
But, he needed differentiation not for first derivatives, but for second derivatives. His grand discovery was this: the laws of motion become clear not when we try to explain an object’s position, nor when we try to explain its velocity, but only when we try to explain its acceleration!
So, just as Eilenberg and Mac Lane needed categories to define functors, and needed functors to define natural transformations, so they could make precise sense of cohomology theories being ‘naturally isomorphic’…
… Newton needed position to define velocity, and velocity to define acceleration, so he could make precise sense of
There’s even a nice analogy between these two trios. An category is a place for an object to sit. A functor is a way of changing an object. A natural transformation changes a way of changing an object.
I haven’t quite figured out how to exploit this analogy. It’s interesting to note that an -category internal to is automatically a chain complex and is thus related to second derivatives by the formula
But somehow I should get natural transformations (chain homotopies) into the game.
Anyway, it might be a sign of genius for somebody to study a trio of concepts, each systematically derived from the previous one, where the third one is the charm.
This is one of the few times where my not actually knowing any mathematics would have helped me answer a question on the n-Category Café! Unfortunately for me, I didn’t notice the thread until the question had already been answered.
I haven’t quite figured out how to exploit this analogy.
This sounds like another challenge :)
Is there some way to imagine a derivative as somehow being internal to something?
I’m obviously thinking out loud where “thinking” is probably even a stretch, but going the other way:
1.) acceleration
2.) velocity
3.) position
makes me think of “integration without integration” for some reason…
Hi John,
Maybe there is another kind of generalization for newton’s law that uses a kind of of 3 charm:
“BF-theory is a gauge theory of n-gerbes with connection over n+3-dimensional spacetime S (often, and for larger n always, assumed to be trivial in the existing literature). The gerbe connection provides us, locally, on patches U i⊂S, with an n+1-form”
http://golem.ph.utexas.edu/string/archives/000777.html
See equation 4. Isn’t like taking the equations of movement from that action like getting a kind of generalized newtons law for D-dimensional particles with exotic statistics in a D+3 space? No wonder that is called a motion group. And isn’t the reason that there is a 3 dimensional step related to the fact that the surface in which the motion happens follows d^2=0?
Are there infinitesimal versions of functors and natural transformations that make the analogy better? I can think of one situation: This is for fiber functors on Tannakian categories. Then the invertible natural transformations form a group scheme, so that one can look at its Lie algebra.
But are there some general notions?
I haven’t quite figured out how to exploit this analogy.
Actually you have! There was some article we once wrote:
let be a Lie group – and let’s concentrate here on to get closest to the Newtonian example regarded as a 1-object groupoid .
Functors from to itself and natural transformations between these arrange themselves into the smooth 2-groupoid .
To find the infinitesimal version of this triple gadget (object = position, morphism = functor, 2-morphism = natural transformation) we proceed the way one differentiates any Lie group: we map smooth 1- and 2-parameter paths into this:
So let be the strict 2-groupoid of smooth paths in the plane and consider smooth 2-functors
These 2-functors are in bijection with 1-forms on .
The way the equivalence works is precisely the Newtonian example:
Regarding the value of the functor on a path in as a trajectory in , gives the velocity of this trajectory with respect to “parameter time”. For instance if you think of the standard speed parameterized path in and write , then is precisely the velocity of the trajectory.
And the value of the 2-functor on 2-morphisms then, i.e. its value in natural transformations of functors on , is indeed given by the (“exterior”) acceleration .
This example, incidentally, when we pass from to more general groups and include linear representations, then also subsumes the first Trimblean triple as a “special case” of “category-functor-natural transformation”.
It is funny how you can write something that looks like english and makes us feel as if it should have been obvious.
I WISH I understood what you just said because it sounds neat, but I really have no clue :) The failing is entirely mine, but reading this gave me a smile this morning (which is rare these days)
It is funny how you can write something that looks like english and makes us feel as if it should have been obvious.
What is also a bit funny is that I have been talking about this here a lot, with the increasing impression of having pretty much decoupled from my suroundings thereby in an unfortunate way.
This analogy
has occupied me quite a bit in my attempts to find the right abstract way of looking at curvature in nonabelian differential cohomology. It led me to those “tangent categories” (which yield the appearing above) and “arrow theoretic differentials” etc, which you may remember, as for instance from page 10 here. Cleaned-up bits of that are in my articles with Konrad Waldorf.
By now I have a more, say, high-brow way of thinking about curvature things, which comes back to language that more people may be able to connect to at least vaguely, but it is less close to the above analogy.
If is a 1-form on given by
then
Should we throw a Hodge star in there so that
and
where is acceleration?
To improve the analogy
it is good to think of movement of relativistic particles moving at speed of light: for them all acceleration is always transversal and given by a bivector.
Indeed, for a charged relativistic particle the above analogy becomes an honest duality: the connection (1-form) of the background field pairs with the velocity (1-vector) of the particle, and the curvature (2-form) of the background field with the acceleration (bivector).
In the “Geometric Algebra” corner they are good at emphasizing this point that acceleration is really a bivector (in relativistc mechanics) and that the Lorentz force law (which is Newton’s law for the charged particle!) just equates the acceleration bivector with the curvature of the background field – let’s see if I find a good reference to point to… ah, maybe slide 6 here.
it is good to think of movement of relativistic particles moving at speed of light: for them all acceleration is always transversal and given by a bivector.
I happen to think that this situation applies to all particles in nature, i.e. instantaneous velocity is always the speed of light, but average velocity depends on the frequency of changes in direction (which depends on mass via ideas similar to the Feynman checkerboard). BUT that is another story :)
I happen to think that this situation applies to all particles in nature, i.e. instantaneous velocity is always the speed of light, but average velocity depends on the frequency of changes in direction
Sure. Fundamentally, all particles are massless.
Only because some of them get kicked around by the Higgs background field a), b), c) don’t they quite manage to go straight at the speed of light and effectively seem to go straight at a lower speed f).
So any question concerning physics at its fundamental level (and I suppose that’s what we are interested in here) should concentrate on massless particles.
Neat. Although it probably should have been obvious if I were paying attention, I never thought about Higgs this way before. My formal field theory education ended with Sakurai in 1992 :)
Hi people,
Doesn’t this 3 charm look more natural in the discussions of the Motion Group in terms of BF-Theory?
This post at string coffe table seems to put BF theory in relation to this 3 charm in a more motivational and interesting position, that is, leading to more richer and genercial (anyon or nonabelion) structures. It seems that BF is “critical” in relation to 3 charm.
I am sorry, but I can’t see the point in discussing so much classical theories with such nice ideas and formalisma. It is like trying to kill an ant with a nuclear bomb. Wouldn’t it a much better use of them for more “advanced” theories, like branes, topological strings and LQG?
I say that, because in that Urs’s post, the Motion Group is related to this things.
Well, maybe I am ignorant, and for all of you, when you talk about classical theories, everything that is more advanced is obvious for you guys.
So, just as a curiosity, how, or would it be possible, that anything that is non classic could fit in there? Hall Effects and branes, for example…
Daniel wrote something like:
I am sorry, but I can’t see the point in discussing so much classical theories with such nice ideas and formalisms. It is like trying to kill an ant with a nuclear bomb. Wouldn’t it a much better use of them for more “advanced” theories, like branes, topological strings and LQG?
I disagree.
I want to get to the bottom of things. So, if a mathematical idea offers new viewpoints on physics, I want to start by thinking about the simplest situations where these new viewpoints apply. If the idea only applies to fancy “advanced” theories, then that’s what I’ll think about. But if the idea already sheds new light on something simpler, like classical mechanics, then that’s where I want to start.
The professor for the only “physics” class I took in “college” (there’s an unimportant meaning to each set of scare-quotes) once told me that the first question about any new model is “where is the hydrogen atom?” I’d wager that by now “where is the simple harmonic oscillator?” should come first, if not “where is the free particle?”
Here is a the way a kindergartener might think about it…
We want to be a velocity 1-form, but a velocity should probably come from a position 0-form . An obvious thing to do given a position 0-form is to take its exterior derivative
A particle will follow a path
so that pulling back we get
I don’t think we want to set though because the curvature would be zero. So instead, maybe we want
If that is the case, then the curvature
is the (Hodge star of the) Laplace-Beltrami operator applied to the position one form. I’m in no position to check, but I would guess that
where is the acceleration.
Then if I were to further guess, I’d guess that
which would mean
I warned that this was kindergarten math, but it’d help to understand it in the more traditional language and then see how to relate that to the groupoid version.
I like this example very much and think it could help illustrate a lot of neat concepts that you guys are working on.
If we were to work through the relativistic charged particle in an electomagnetic field in detail, we would rather need to think of paths in a directed space.
If we were to work through the relativistic charged particle in an electomagnetic field in detail, we would rather need to think of paths in a directed space.
Now that you say this in the present context, it made me have an idea:
Let be a globally hyperbolic Lorentzian manifold.
Suppose we have an electromagnatic background field on , given by a smooth parallel transport functor on all paths in .
Then for a charged particle zipping through , what equals its acceleration bivector is not the full curvature bivector of , but just its electric part, which is its projction onto the timelike direction.
(Still see the equation on slide 6 here).
So, how can we understand this functorially? Well, let’s follow your suggestion and restrict our parallel transport functor to (future-)directed paths, i.e. to those paths which correspond to morphisms in the poset structure on .
This gives us a category of future-directed paths (being a subcategory of ). So we can restrict our background field
along the inclusion and consider the curvature of tis . I’d need to think about this more in quiet, but it seems that the curvature of would be precisely the 2-form/bivector which equals the acceleratoin bivector in the Newton-Lorentz force law.
But let me think about this. First of all I haven’t before defined curvature of a functor which is not a functor on a groupoid of paths. But roughly the above looks like it might be a useful way of looking at things here.
Yeah. This example, although already pretty simple, may not even be the simplest.
An even “simpler” example (for undergrads) might be to consider a Galilean spacetime as a directed space. Seeing how Newton’s laws fall out of groupoids would be very neat. Seeing acceleration as curvature would also make a nice segue to general relativity. In fact, I suspect that acceleration can be seen as curvature in all three frameworks
* Netwonian
* Special relativistic
* General relativistic
That would be a great way to teach this stuff.
Note: I think the Lorentzian version is actually “simpler” than the Galilean version, but undergrads may not be ready for that (depending on background).
It’s been a while since I wrote down the Laplace-Beltrami operator, but since you’re using a connection , I should probably use instead so
where is acceleration and is the volume form on .
John wrote:
I haven’t quite figured out how to exploit this analogy.
Urs wrote:
Actually you have! There was some article we once wrote…
Hmm. Maybe you’re right. At least you’re getting close, but I still don’t feel like I see a perfectly beautiful example of a situation where:
Working out this analogy in the most beautiful way in the simplest example would be a very good thing. It might help some people understand category theory and other people understand physics.
To improve the analogy
it is good to think of movement of relativistic particles moving at speed of light: for them all acceleration is always transversal and given by a bivector.
I’ll have to think about that more…
What is also a bit funny is that I have been talking about this here a lot, with the increasing impression of having pretty much decoupled from my suroundings thereby in an unfortunate way.
You spend a lot more time thinking about math and physics than explaining it (and thinking about how to explain it), and you’re smart, so you go faster along your own particular line of research than anyone can keep up with you.
I spend more time explaining math and physics than actually thinking about it, so I don’t have this particular problem.
With luck, someday you’ll be famous enough that a bunch of people will spend their full time trying to understand you and develop your ideas further. This seems to be a key element of Witten’s and Connes’ success… though of course they also do spend a lot of time giving talks and writing books and trying in other ways to make their ideas easy to follow.
The way the -Category Café works, you have a lot of people reading what you write… but it’s a blog, so if I’m a typical example, a lot of these people start skimming whenever it gets too hard to understand.
Okay, let me try to find a setup in which this becomes very obvious.
First of all, the more obvious part of the analogy is
so let me start with that.
Here a simple situation is that where
- each fiber looks like a group , regarded as a one-object groupoid (hence a category), which I write as ;
- all functors go from to itself;
- and the natural transformations in question are the natural transformations between these functors.
And we want to assume that is a Lie group, so that we can proceed with the analogy which clearly presupposes a smooth setup.
In this case, the main subtlety in making sense of the above analogy comes from the existence of outer automorphisms on . So, in the spirit of making everything as clear as possible, let me assume a situation where these vanish. For instance assume (i.e. the compact real form of ).
Then (I am saying this to be self contained for the record and for other readers) the relation between the two lines in the above analogy is established by the notioin of smooth 2-functors
from the 2-groupoid whose
- objects are points in a manifold ;
- morphisms are classes of smooth paths in ;
- 2-morphisms are classes of smooth surfaces in .
Assume this 2-functor assigns
1) to each point the fiber of a -principal bundle over , which we can locally identify with and hence with the category
2) to each path a functor from the fiber over to the fiber over .
Theorem: such smooth assignments of functors to paths are in bijection with connections on the -principal bundle: upon identifying the fibers with the functor is given by an automorphism of hence, by assumption, by an element of , and this element is the parallel transport .
3) to each surface a natural transformation of functors . Since has a single object, such natural transformations are given by elements of once we identify the fibers with .
Theorem: this element of is the group element where is the curvature 2-form of and denotes the surface ordered exponential integral of a 2-form with respect to (as appearing in the nonabelian Stokes theorem).
So that’s how this works. Before going into the further analogy
let’s massage the above kind of observation into a form that we are all happy with.
Using the symbol BG in that way is confusing to us of an earlier generation.
Using the symbol BG in that way is confusing to us of an earlier generation.
It’s actually meant to be maximally non-confusing and adapted to the needs of the earlier generation, because everything you do with the groupoid remains true after geometric realizatoin for the space .
Please see our long discussion about this at the end of Lab: category algebra.
I think I can sympathize with what Jim is saying, because it’s taken me a while to get used to the notation as you are using it here. I was brought up thinking that is a space, not a 1-object category. And old habits die hard.
The philosophy, if I understand it, seems to be: we should be thinking of the one-object category , or the space , as incarnations (in different categories) of the same basic concept, and so one is within rights to use the same notation for each.
But, this philosophy is in some conflict with the philosophy that a concept (e.g., the concept of the real number system) is never in isolation, but is always relative to some specified categorical context in which the concept is regarded as living (e.g., real numbers as topological group, real numbers as discrete field, and so on). In other words, the trouble with is that most mathematicians will probably reflexively think, “ lives in .” Followed by, “no wait, that doesn’t make sense here.”
I don’t recall whether Urs uses different notation for the topological sense of , but all I’m saying is that the topological meaning is probably the default meaning of for most mathematicians. I think I’m at the point where I don’t mind Urs’s way of using for the one-object groupoid, but for me it took some getting used to, some additional mental training, and that may be the added difficulty that Jim is referring to.
I have to agree with Jim and Todd, here. To a topologist, is a functor from categories to spaces, and means either the group or the one object category.
Because of my short attention span, I haven’t checked if someone has already pointed this out, but there is a stack-theoretic reason for using Urs’ convention.
One can define the stack BG in various reasonable settings by taking its sections BG(X) on some object X to be the groupoid of principal G-bundles on X. The notation here seems to be perfectly consistent with the occurrence of BG in homotopy theory.
But then, it’s straightforward to check (say for usual topological spaces, or schemes with the etale toplogy) that BG as a stack is represented by the groupoid BG under discussion, that is, with one object and automorphism group G, thought of as `the point mod G.’
taking its sections BG(X) on some object X to be the groupoid of principal G-bundles on X.
so in the top context BG(X) = [X,BG]
where [ , ] = homotopy classes of maps?
would that be a definition or a theorem?
an important to me distinction
Sorry about the confusing formulation. Let’s define the stack by taking its sections on to be the groupoid of principal bundles on . So the objects are principal -bundles and the morphisms are isomorphisms. On the other hand, consider the groupoid with one object having automorphism group . Then it is a (small) theorem that represents .
To spell out a bit what this means, recall that any groupoid, say in topological spaces,
(sorry, there should be two arrows, but I’ve forgotten how to do that in latex)
determines a pre-stack, whose sections on a space are the groupoid whose objects are and whose morphisms are . There is a rather obvious notion of `stackification’ of a pre-stack, which I’ll refrain from formulating for now. Then the theorem is that the stackfication of is exactly . Of course if you think about it a bit, this theorem is essentially the definition of a principal bundle. So maybe the original assertion is a definition after all.
I’m happier with `represents’ rather than `is’
But, this philosophy is in some conflict with the philosophy that a concept (e.g., the concept of the real number system) is never in isolation, but is always relative to some specified categorical context in which the concept is regarded as living.
Of course, as you probably know, the argument is that Gpd should be considered as a full sub--category of Top, so that as a groupoid and as a space really are the same object. Geometric realization from groupoids to spaces (as opposed to from categories to spaces) is, homotopically speaking, a no-op.
I also want to point out that actually has two different meanings in classical topology:
I suppose the intuition behind this conflation of notation is that if you consider a group as a one-object groupoid and take the realization of its nerve, you get the same thing as if you consider it as a discrete topological group and take its classifying space.
But the point, for me, is that these two meanings are actually in conflict. For example, if is a topological group then . But if is a groupoid, then should clearly mean its set of isomorphism classes of objects, and should clearly mean the automorphism group of , but these are equal to and of the realization of the nerve of . Likewise, the inverse of the delooping is the loop space, , but the realization of the nerve has no inverse for categories, and for groupoids its inverse is the fundamental groupoid , again a no-op on 1-types.
Or suppose that is a monoidal groupoid. I can take the nerve of its classifying space to get a topological monoid (or -space if it is not strict monoidal), and then I can deloop that monoid. In classical notation I would have to write the result as , where the two s denote completely different operations!
I do realize that it is difficult for people to get used to changes in notation. But I think that using to mean both (a) the delooping of a topological group and (b) the one-object groupoid associated to a group is less confusing than using it for the two classical meanings, because (a) and (b) are much more closely related (in fact, essentially the same, modulo the equivalence of groupoids with 1-types). It took me quite a while to get over my confusion about the two classical uses of , which is one reason I am on such a crusade to expunge its use for the realization of nerves.
I also feel that the identification of groups with one-object groupoids does, overall, more harm than good—more harm than the identification of groupoids with 1-types! (Groups can instead be identified with pointed connected groupoids.) So I think it is really important to have a notation for passage from groups to one-object groupoids, to remind us that there really is something going on.
I think that someone, John maybe, has used ordinary for the delooping of a topological group and script for the one-object groupoid associated to a group. If that makes you less confused, I would be happy to use that.
Mike Shulman wrote:
Of course, as you probably know,
Not me - as I say, consider your audience
the argument is that Gpd should be considered as a full sub-(infty,1)-category of Top, so that BG as a groupoid and BG as a space really are the same object. Geometric realization from groupoids to spaces (as opposed to from categories to spaces) is, homotopically speaking, a no-op.
What’s a no-op?
I also want to point out that B actually has two different meanings in classical topology:
1. If G is a topological monoid (or an A_infty-space), then BG is its classifying space, aka “delooping”.
2. If C is a category, then BC is the realization of its nerve.
I suppose the intuition behind this conflation of notation is that if you consider a group as a one-object groupoid and take the realization of its nerve, you get the same thing as if you consider it as a discrete topological group and take its classifying space.
YES!
but the realization of the nerve has no inverse for categories,
how about the singular complex?
perhaps only up to infty-homotopy
Or suppose that C is a monoidal groupoid. I can take the nerve of its classifying space to get a topological monoid (or A_infty-space if it is not strict monoidal), and then I can deloop that monoid. In classical notation I would have to write the result as B(BC), where the two Bs denote completely different operations!
am I the only one confused by that - the nerve of a space? and completely different?? in what categories?
So I think it is really important to have a notation for passage from groups to one-object groupoids, to remind us that there really is something going on.
That I’d agree with.
I think that someone, John maybe, has used ordinary B for the delooping of a topological group and script B for the one-object groupoid associated to a group. If that makes you less confused, I would be happy to use that.
Something like that but don’t depend on fonts.
Of course, as you probably know,
Not me - as I say, consider your audience
Sorry, “you” in that sentence was addressed to Toby, not Jim, since it was his specific comment that I was responding to. I wouldn’t expect a classical topologist to know “that Gpd should be considered as a full sub-(infty,1)-category of Top”!
What’s a no-op?
Means “does nothing;” from computer science. I refer to passing across an equivalence of categories, or an equivalence of homotopy theories, as a no-op because no information is lost or gained; we only change our point of view on some “real” underlying object. In this case, the relevant fact is that the homotopy theory of groupoids is equivalent to the homotopy theory of 1-types.
but the realization of the nerve has no inverse for categories,
how about the singular complex?
perhaps only up to infty-homotopy
No, a category cannot be recovered up to equivalence from the singular complex of its nerve. There is a (zigzag of) functors relating a category to the category of simplices of the singular complex of the realization of its nerve, which induces a weak equivalence of nerves, but I don’t think it is reasonable to view that as recovering the category itself. Equivalence of nerves is not the fundamental notion of “sameness” for categories.
Or suppose that C is a monoidal groupoid. I can take the nerve of its classifying space to get a topological monoid (or A_infty-space if it is not strict monoidal), and then I can deloop that monoid. In classical notation I would have to write the result as B(BC), where the two Bs denote completely different operations!
am I the only one confused by that - the nerve of a space? and completely different?? in what categories?
Sorry, where I said “nerve of its classifying space” in that paragraph I should have said “realization of its nerve.”
Sorry, “you” in that sentence was addressed to Toby, not Jim, since it was his specific comment that I was responding to.
You mean Todd, not Toby. You already convinced me back on the Lab!
Sorry! I knew who I meant… but apparently my fingers didn’t.
A couple of comments on the discussion about the “B G”-notation:
1) actually I am typing
“mathbf{B}G”
to retain an indication that this is closely related to but different from the topological . You can see this in my LaTeX notes. But for some reason, lately the boldface math seems not to be displayed here anymore (??)
2) You may recall that we had this discussion about a year ago already. Back then I used to write and “you” (people here) convinced me that it would be better to use .
3) and I was happy with this: as Todd and Minhyong mentioned: when writing most statement you make just literally remain true whether you interpret them
- in groupoids
- in simplicial sets
- in stacks
- in topological spaces.
- etc.
4) It seems there is no globally consistent notation that fits every single convntion.
5) finally: I claim we need some suggestive notation to distinguish the discrete monoidal groupoid from its one-object groupoid incarnation.
6) of course we can always fall back to writing .
It’s unfortunate that time and pixels were wasted over a bug in the itex2mml code.
It seems that “\mathbf{B}” does not work because it is expecting more than one character (?). For example, “\mathbf{BE}” does work. I’m trying to find a hack to work around it, but hopefully someone somewhere can fix the bug.
For more on this problem with \mathbf{}, see my comment over at TeXnical Issues. We should discuss such problems there, not here.
Since we’re all expressing our feelings about the notion for ‘the group regarded as a one-object groupoid’, I can’t resist joining in.
While I’ve occasionally teased Urs about making a big deal about the distinction between groups and one-object groupoids, there are certainly times when this distinction must be made clear. I think the notation is good for this — although it needs to be explained every time one starts talking to a new person!
In fact, Jim Dolan and I advocated this notation back in 1998, on page 13 of our paper on categorification. We listed a bunch of ways to hop around the periodic table, and this was one:
Delooping: given a -tuply monoidal -category, there should be a -tuply monoidal -category with one object obtained by reindexing, the -morphisms of being the -morphisms of . We use the notation ‘’ and call the delooping of because of its relation to the classifying space construction in topology.
Right now we’re talking about the special case , — and the special case of that where our monoid is actually a group.
Anyone puzzled about the relation between this concept of ‘’ and another construction with the same name — the classifying space of a topological group — should check out page 22 of the same paper.
I agree with Mike that the nerve of a category deserves a different name. Not being very creative, I use .
John, Mike,
thanks for the comments on ! I am glad we are finally discussing this properly and converging to a joint attitude.
Here is one aspect which one might add to the discussion (but please check critically what I say now):
I am thinking that we have the operations of looping and delooping whenever we are in the context of a category with interval object ( Lan) .
Given any pointed object in such a category, we always have a notion of its loop object , defined as the pullback which always inherits an monoidal structure. So we can then ask for an object with an essentially unique point such that .
Maybe we should write an nLab page called “on the uses of B” or something.
Your notion of loop object in a category with interval object sounds reasonable. Another general notion of loop object is the following homotopy pullback: As pointed out by Lurie, from this point of view the existence of deloopings is a type of -categorical exactness. The loop object , or more accurately the groupoid , is the kernel of the point . Then if you have any internal group(oid) , exactness says that it is a kernel, i.e. that for some .
again equiv as?
also true for a more general class of monoids but not all
Well, classically, as I’m sure that you (Jim) know (since you helped invent the subject), but other readers might not, one has the equivalence for any “grouplike -space.” In an exact -category one has the equivalence for any “internal group(oid) object;” this is made precise in Lurie’s work but amounts to about the same thing: a space that has a multiplication which is associative, unital, and has inverses up to all higher homotopies. (Formally, Lurie’s notion is more akin to a Segal space than to the operadic approach.) In a grouplike -space, the inverses are only stipulated to exist up to homotopy rather than all higher homotopies, but one of the special things about inverses, like adjoints, is that they can always be “boosted up” to become coherent.
Another general notion of loop object is the following homotopy pullback
Good that you mention this, because it gives me an excuse to raise again an issue which I tried to discuss recently in the Ben-Zvi thread:
first, notice that the Lab already essentially mentions this example (but I’ll make it more explcit):
at generalized universal bundle it says that the fiber of the universal -bundle (for pointed as in your comment above) is . This is computed by the pullback
and this happens to be a fibrant replacement of the diagram
and hence computes the homotopy limit (as described there in detail) if is an un-directed object (i.e. a groupoidal object), i.e. if is a path object: in this case we have and is an (-)group.
But for many applications it is important to speak of in the caser that all this does not hold, but where still exists as an ()monoid.
For instance for with ground field as the point we do not have and (using my definition above) is not a group but is the monoid . Similarly for with as a point, in which case is , etc.
In all these situations the definition above still works, but the one in terms of homotopy limits does not.
It is these kind of examples (where we are talking about -bundles with not groupoidal) which make me hesitate to pass, for instance, to the -category associated with my underlying homotopical category: this forgets the directed homotopies which are important in some applications.
Well, or so I think, maybe I am mixed up about this. I’d be interested in hearing comments.
Well, I would argue that when there are “directed homotopies” in sight, often you aren’t really in an -category but some sort of -category for . In that case the homotopy pullback should be replaced by a comma object. I just wrote a bit about that in my own web at exactness hypothesis.
Well, I would argue that when there are “directed homotopies” in sight, often you aren’t really in an -category […]
Yes, that’s exactly my point. Well, I guess I shouldn’t be arguing here in the context of a discussion that I had with somebody else in some other thread.
But I wanted to amplify this because it felt to be of some relevance in the context of the discussion here. It makes me grow more fond of the concept of a closed monoidal homotopical category with interval object as a good context for higher abstract nonsense.
I just wrote a bit about that in my own web at exactness hypothesis.
Interesting, thanks for the link.
Your comment about the comma object there provokes an obvious question: can we find the open entry in the analogy
?
Well, I suppose this is asking for directed homotopy theory…
One last comment, concerning your remark about as a codescent object:
by the general nonsense of codescent, I think it should be true that the realization of as a strict -category is the coend
(with the th oriental, as you know).
Incidentally, this is (even if not in this language) the motivating example that Lafont, Métayer et al. give at the beginning of a coupel of their articles on strict -categories for motivating the use of (strict) higher categories in computer science.
can we find the open entry in the analogy pseudo-limit : homotopy limit :: comma object : ??
I would personally say “weighted homotopy limit”. The best way I know of to use homotopy theory to get at non-invertible higher morphisms is to use enriched homotopy theory: the homotopy theory builds in the homotopies and the enrichment gives you the directionality.
by the general nonsense of codescent, I think it should be true that the realization of as a strict -category is the coend
Yes, certainly!
I take it [I,X] means Map(I,X)
and not homotopy classes as elsewhere?
That last equiv is as ???
and BOmegaX equiv X? as ???
Note also that if that bottom arrow were the diag X –> X x X
then the pull back is the free loop space more beloved of physicists.
Note also that if that bottom arrow were the diag then the pull back is the free loop space more beloved of physicists.
Sure, this is discussed at
I take it means
Essentially, yes. The thing is that when talking about this object we are (or at least I am in the above discussion) placing ourselves in the context of a category with interval object, where is that interval object, which means in particular that our category is closed monoidal homotopical and hence in particular closed monoidal.
With this context in mind I am following standard convention in enriched category theory to denote the internal hom from to by .
So in particular if nice topological spaces then, yes, .
But if for instance (the other case occuring frequently in our discussion here) then is the functor category.
I didn’t mean Maps to denote topology
Hom would’ve been equally misleading
Perhaps I should have said Mor
Urs wrote a nice explanation, ending with:
So that’s how this works. Before going into the further analogy
let’s massage the above kind of observation into a form that we are all happy with.
I’m very happy with how you described this analogy:
Maybe other people find the path 2-groupoid scary, or smooth 2-functors
but I’m comfy with these concepts, and eager to explain them to anyone who has questions.
So maybe we can focus on the next stage, which is understanding the analogy
I don’t understand this very well! It’s embarrassing, since I’ve spent years thinking about each row.
This thought keep crossing my mind and I’m not sure if it is relevant, but thought I would say it out loud and see what happens…
Given a smooth curve
in a smooth manifold , what if we thought of as a fiber over so that a curve in is a section of the bundle ?
Then “position” is like a tangent vector and “velocity” is like a connection telling you how to parallel transport a tangent vector, i.e. position, from to , where
and
But then I get lost on acceleration and curvature…
Treating as a directed space might help because then curvature/acceleration might come about as a type of Aharanov-Bohm effect, which has a certain poetic aspect to it.
I’m starting to see diamonds, so I better stop their because I’ve already rambled too much.
Ok. When I said I would stop I lied :)
I just started a draft here:
—-
Consider two paths on a directed graph that both begin at the point and end at the point in the figure below
The edges represent velocities and the area enclosed represents the acceleration.
If the acceleration is zero, then the area collapses and any two paths that connect the same two points must correspond to the same straight line, i.e. velocity does not change.
Ok. If I’m going to ramble…
Let’s get ready to ramble!!
Combining three ideas: acceleration as area, Feynman checkerboard, and Higgs, then it is tempting relate the area to mass. If the edges are lightlike, then mass = 0 corresponds to acceleration = 0, so particles would travel at the speed of light.
Massive particles have a nonzero acceleration 2-form.
I’ll state the crux even though at this point it is obvious.
Fiber = Position
Connection = Velocity
Curvature = Acceleration
Coupling this to electromagnetic theory should be trivial too. The minimal principle probably leads to the area 2-form
being replaced with
where is the electromagnetic curvature
Then “zero acceleration” means
There may be a Hodge star or sign missing, but I’m pretty sure the basic idea is correct.
Like the portrayal of Salieri in Amadeus, I feel teased by seeing a flash of something beautiful, but cursed because I do not have the gifts needed to write it down.
I’ve been doodling trying to make some sense of this in simplistic terms, but with little luck. I think I’ve settled on a picture that even a high school student could understand, so I hope it fits into the picture that I’m anxiously waiting to hear about from Urs. Anyway, here is the picture:
———-
Galilean Spacetime
Consider two paths that begin at and end at enclosing the spacetime region depicted below
Since both paths end up at the same point, the velocities must satisfy the relation
which implies that we can write
and
for some constant .
———-
In haste, I wrote some nonsensical equations earlier, but from the picture above, if you think of the velocity as a “holonomy along an edge”, then the expression
says there is no curvature, i.e.
and
Nonetheless, I remain convinced that acceleration is somehow related to a 2-form defined over the 2-dimensional Galilean spacetime region, i.e. 2-diamond, above. This 2-form tells you how to (2-)transport the left path across region to the right path.
It actually seems to fall out nicely in the discrete world, so it should have an acceptable continuum formulation as well.
Another indication, which is also simplistic but meaningful in my opinion, is that
* It takes one tick of a clock (zero intervals) to measure position.
* It takes two ticks of a clock (one interval) to measure velocity.
* It take three ticks of a clock (two intervals) to measure acceleration.
—-
In a directed graph:
* It takes one tick to traverse a node (or 0-diamond)
* It takes two ticks to traverse an edge (or 1-diamond)
* It takes three ticks to traverse a 2-diamond
——
Putting these together:
* Position lives on 0-diamonds and is a 0-form
* Velocity lives on 1-diamonds and is a 1-form
* Acceleration lives on 2-diamonds and is a 2-form
That is about as far as I’ve been able to get so far…
Hi John,
So maybe we can focus on the next stage […]
Okay. In my comment which started this discussion I essentially pointed out that:
the bijection/equivalence
(smooth 1-functors from ) (1-forms)
is – manifestly the way one proves it – controlled by the concept of velocity:
if you think of as 1-dimensional, then the 1-form on with values in is nothing but the gadget which reads in a bit of “coordinate time” for a trajectory on and spits out the velocity tangent vector of that trajectory with respect that coordinate time.
This is really what “parallel transport” of 1-forms means: find the trajectory such that its velocity is at every point prescribed by the given 1-form.
Think in the case of the functor as describing a particle which propagates on a group manifold. The fact that the functor is smooth says that we have a smooth trajectory. The fact that it is functorial means that this trajectory is obtained by intergrating a velocity 1-form.
Now, in my comment which started this discussion I next said that on 2-cells our parallel transport computes the “exterior acceleration”: given a 1-parameter family of trajectories, the 2-form that controls the value of a smooth 2-functor on it is the exterior differential of the “velocity 1-form”.
At first it is admittedly a bit of a stretch as far as the Newtonmian concept of acceleration goes to call this curvature 2-form the “acceleration” here, though it does consist of second derivatives of the original trajectories.
But then notice the Newton-force law for charged relativitic particles: the Lorentz-force law. it says that the accelreation bivector of the relativistic particle equals the (electric component of) the curvature 2-form. So we have
There must be better ways to say this. But maybe this helps to indicate what I am thinking of.
I was playing around with this a little last night…
Let denote the 4-velocity of a charged particle and denote the corresponding dual 1-form, the acceleration 1-form may be written as
where is the curvature 2-form and the “acceleration 2-form” (corresponding to the acceleration bivector) is
I guess this is equal to . To see this, note that
We have
and
so that
I believe this is the clue that Urs mentioned, but made a bit more explicit. On the left is curvature and the right is the acceleration 2-form.
In my Galilean spacetime example, I think it is the “acceleration 2-form” that specifies the 2-transport even in the absence of an electromagentic field.
Eric wrote:
Hmm. Before you guys were saying something really nice happens when we assume our particle is massless. Then its velocity vector is lightlike, which gives
Your calculation seems to work better for a massive particle, where we can use the metric to turn its velocity vector into a 1-form with . Of course we can always find some 1-form with , but probably not in a canonical way (using the metric) if is lightlike.
I need to think about this more… there should indeed be something fun going on here, and Urs has probably explained it already, but I need to unpack it and ponder it.
In my Galilean spacetime example, I think it is the “acceleration 2-form” that specifies the 2-transport even in the absence of an electromagentic field.
If you take some light whose color is magenta, is that an example of an electromagentic field?
(Sorry, I’m just a goofball.)
You can rest assured knowing that when I start writing down equations, a good 10-15% of them will not be correct. That is a small price to pay if we can push the idea along in my opinion :)
I don’t think
but it was a convenient mistake to illustrate a partial relationship between curvature and the acceleration 2-form.
As far as massive vs massless particles, one of the points (I think) is that for massless particles, the acceleration is zero so they will always propagate in a straight line. Bringing Higgs in would be fun, but then I think the definition of “mass” becomes tricky.
If there is any truth to anything I’m saying, it is only because I’m trying to parse what Urs already said in a way I might understand. There is a good chance I am just spewing a bunch of nonsense though (it wouldn’t be the first time!).
I hope the spacetime diagram ends up being useful because it is very pretty.
Before you guys were saying something really nice happens when we assume our particle is massless.
I may have to take this back, sorry. What I really argued so far is that the nice 1-2-3-sequence we are looking for is obtained when thinking of acceleration for relativistic particles, which is naturally a bivector that naturally pairs with the curvature 2-form of the background field which exerts the force.
Above I thouhgt lightlike particles would make things even nicer, but possibly I was being silly. Let me think about it a bit more…
Oh.
When there is no electric field, then and the acceleration, velocity, and magnetic field are mutually orthogonal. In this case, the geometric product
is the same as the wedge product and you can “invert” the interior product so (I think)
In this case, you have
and the relation between curvature and the acceleration 2-form is explicit.
I think the math is correct this time. To check
since and . Also
which is the Lorentz force law.
Is that what you’ve been saying all along? :)
I’m getting closer. Promise. I think what I did above is a special case of what you’ve been saying.
Since
it is generally true that since is nilpotent, i.e.
since is antisymmetric. Therefore, the acceleration 2-form is always the same as the geometric product of velocity and acceleration
At least when the acceleration comes from a curvature.
SOOO…
When the acceleration comes from some curvature, then the acceleration 2-form is “special”. I think it is special because it specifies the 2-transport between particle world lines.
Help me Mozart!
When the acceleration comes from some curvature, then the acceleration 2-form is “special”. I think it is special because it specifies the 2-transport between particle world lines.
Does that paragraph make anyone else think about general relativity?
I wish I was smarter!
With that last comment in mind and going back to the 2-diamond example:
If we switch to a relativistic perspective and think of the edges as being lightlike (geodesics), then this depicts a discrete general relativistic picture where the two paths are actually geodesics implying the spacetime surface contains curvature and the acceleration 2-form specifies the 2-transport from the left world line to the right world line.
I had a “Eureka!!” moment.
In the spacetime diagram
I was trying to figure out exactly what should go on the edges and what should go on the surface. I knew the edges had to do with velocity and the surface had to do with the acceleration bivector.
Then I got it (by following the link the Urs gave to the geometric algebra stuff!). I had the 2-arrow pointed in the wrong direction. Instead of 2-transporting the left worldline to the right worldine, it should actually be a “boost” along the time direction, i.e. the 2-arrow should be “up”!
But now what gets assigned the edges are “rotors” from geometric algebra. The edge rotors represent a Lorentzian “boost” along the edge direction. The surface rotor represents a rotation of the initial velocity edge into the final velocity edge.
If we think of this this as an infinitessimal region, then globbing these things together amounts to intergration. The velocity and acceleration rotors are constant on the infinitessimal 2-diamond.
That is neat!
A little more explicitly…
Almost there.
By the way, I modified the spacetime diagram to make it look more like a strict 2-category at the nLab:
Now, I’m hoping to do something along the lines of explicitly constructing a 2-transport functor
somehow, where is the spacetime algebra.
If things work out, it looks like we might be able to express “surface ordered surface integrals” via geometric algebra.
where is the spacetime algebra.
This “spacetime algebra” is just the Clifford algebra . This is not a 2-category, so your proposed to functor does not quite make sense as stated, as its target should be some 2-category.
Now itself is not a 2-category. But it can naturally be regarded as being an object in a 2-category: namely the 2-category !
if you conmsider that, then the situation you indicate starts to look very much like the situation discussed on page 25 of AQFT from -functorial QFT (#) (the corrected version to appear in print I’ll upload in a few days.)
Thanks Urs. I know you are traveling and very busy (as usual), so I’m slogging along trying to make some progress. When things return to normal levels of insanity, I’m hoping I make enough progress that you can help clean up any remaining issues with just a few brush strokes.
I’ll try to understand the “more correct” object I think I’m after:
where is the strict 2-category representation of the diamond spacetime region . That sounds pretty.
I’m guessing that this is just some extremely simple situation of the generally picture you’ve already written down, but making it explicit at this elementary level would be helpful I think.
I just noticed I read that wrong. I need to learn about .
I know it is probably extreme overkill, but is there a way to think of complex numbers as sitting inside somehow?
Eric wrote:
I know it is probably extreme overkill, but is there a way to think of complex numbers as sitting inside somehow?
Sure! is the unit object of the unit object of .
Here’s what I mean:
is a monoidal bicategory, i.e. there’s a tensor product of -modules.
The unit object for this tensor product is . In other words, you can tensor any -module with and get a -module equivalent to .
is a monoidal category: i.e. there’s a tensor product of vector spaces (which are just -modules).
The unit object for this tensor product is . In other words, you can tenso rany -module with and get a -module isomorphic to .
I hope you see, from the singsong repetitive quality of this comment, that this is just the beginning of a pattern that continues forever.
By the way, speaking of patterns that continue forever: when you keep commenting on your own comments to your own comments, it creates a highly nested thread that’s not much fun to read. All those vertical lines on the left get a bit dizzying! To avoid them, it’s better to keep commenting on the topmost relevant comment.
Eric asked:
is there a way to think of complex numbers as sitting inside somehow?
John replied:
Sure! is the unit object of the unit object of .
Maybe Eric is coming from my statement above that the Cliffford algebra can be regarded as an object of . In that case, maybe he is asking if the complex numbers regarded as an algebra are also an object of . (?)
In this case the answer is Yes, but this is another way we can think of the symbol as naturally denoting something “inside” :
every algebra defines an object of . We can think of these algebras
- as being the objects of (if we adopt the point of view of -enriched category theory in which case the objects of are -enriched categories which are algebras if they have a single object)
- or as being placeholders for the true objects of (namely if we regard this as the bicategory of -module categories), in which case the true object is the category of modules over the algebra (vector spaces on which the algebra is represented!).
The complex numbers regarded as an algebra are an object of in the first sense. As such they may be regarded as representing the category of representations (= modules) of the algebra of complex numbers. But every complex vector space is tautologically such a representation. So . This leads to the perspective that John described as we can find sitting inside its own category of representations
I wonder if it is a coincidence that the functor
on Page 4 looks like a rotation in geometric algebra for any multivector ?
I wonder if it is a coincidence…
I don’t know if one should call it a coincidence or not, but in any case it is not supposed to be a deep. You should think of the Heisenberg picture of quantum mechanics, where time evolution is given by an inner automorphism, i.e. by conjugation with a unitary operator. Also rotations in quantum mechanics are represented by such conjugations on other operators. Whether we have a spin representation or not.
John wrote:
The way the n-Category Café works, you have a lot of people reading what you write; but it’s a blog, so if I’m a typical example, a lot of these people start skimming whenever it gets too hard to understand.
I would say:
The way the n-Category Café works, you have NOT a lot of people reading what you write; IN COMPARISON TO YOUR POTENTIAL AUDIENCE. YOU DO GIVE TALKS BUT PUBLICATION MIGHT HELP. ;-) but it’s a blog, so if I’m a typical example, a lot of these people CAN’T KEEP UP.
It’s like those physics talks in which the slides flash by at the speed of light.
Jim wrote:
The way the n-Category Café works, you have NOT a lot of people reading what you write, IN COMPARISON TO YOUR POTENTIAL AUDIENCE.
It’s hard to know how many people are really paying attention to this blog, but it’s a lot more than the number of people who post comments. It’s not a substitute for putting papers on the arXiv and then publishing them, but it’s a good supplement.
54,902 people have visited the main page of the n-Café since September 25th, 2008. And they come from all over:
United States (US) 18,782
United Kingdom (GB) 6,268
Germany (DE) 4,493
France (FR) 2,271
Canada (CA) 2,163
Australia (AU) 1,471
Netherlands (NL) 1,015
Japan (JP) 1,006
Switzerland (CH) 894
Sweden (SE) 806
Spain (ES) 774
Brazil (BR) 724
Belgium (BE) 649
Portugal (PT) 640
Italy (IT) 498
Poland (PL) 443
Finland (FI) 426
Norway (NO) 421
Austria (AT) 363
China (CN) 352
New Zealand (NZ) 337
Russian Federation (RU) 320
Denmark (DK) 293
India (IN) 284
Greece (GR) 277
Czech Republic (CZ) 222
Argentina (AR) 210
Israel (IL) 205
Bulgaria (BG) 176
Korea, Republic of (KR) 158
Ireland (IE) 155
Croatia (HR) 149
Luxembourg (LU) 146
Thailand (TH) 140
Taiwan (TW) 132
Mexico (MX) 128
Colombia (CO) 122
Singapore (SG) 99
Malaysia (MY) 93
Hungary (HU) 82
Hong Kong (HK) 77
Turkey (TR) 62
Estonia (EE) 62
Chile (CL) 60
Romania (RO) 58
Europe (EU) 48
Philippines (PH) 48
Slovenia (SI) 47
Vietnam (VN) 43
South Africa (ZA) 42
Algeria (DZ) 31
Egypt (EG) 26
Iran, Islamic Republic of (IR) 25
Saudi Arabia (SA) 25
Ukraine (UA) 24
United Arab Emirates (AE) 24
Iceland (IS) 22
Indonesia (ID) 21
Venezuela (VE) 15
Lithuania (LT) 13
Uruguay (UY) 7
Costa Rica (CR) 7
Latvia (LV) 7
Asia/Pacific Region (AP) 7
Serbia (RS) 6
Slovakia (SK) 6
Jamaica (JM) 6
Oman (OM) 5
Nepal (NP) 5
Peru (PE) 5
Jordan (JO) 4
Pakistan (PK) 4
Qatar (QA) 3
Belarus (BY) 3
Moldova, Republic of (MD) 3
Libyan Arab Jamahiriya (LY) 2
Ecuador (EC) 2
Bangladesh (BD) 2
Kuwait (KW) 2
Cyprus (CY) 2
Palestinian Territory (PS) 2
Bahrain (BH) 2
Tunisia (TN) 2
Bolivia (BO) 1
Rwanda (RW) 1
Paraguay (PY) 1
Ethiopia (ET) 1
Guam (GU) 1
Nicaragua (NI) 1
Guatemala (GT) 1
Cuba (CU) 1
Antigua and Barbuda (AG) 1
Dominican Republic (DO) 1
Iraq (IQ) 1
Syrian Arab Republic (SY) 1
Georgia (GE) 1
Bosnia and Herzegovina (BA) 1
Azerbaijan (AZ) 1
Yemen (YE) 1
Malta (MT) 1
Brunei Darussalam (BN) 1
Impressive
but are those people or hits?
The precise way those numbers are computed can be seen here.
Briefly: these numbers count ‘visits’. Each day you go to this blog’s main page counts as a ‘visit’. If you go to main page more than once in a day, that still counts as a single ‘visit’. And if you don’t go to the main page, that counts as no visits at all.
So those of us who never go to the main page, but click through directly from the RSS feed to an individual blog entry, aren’t counted at all?
I believe you’re not counted. I’m getting my data from that ClustrMaps thing on the main page of this blog, and I think it only counts people who visit that specific page.
So, if lots of people use RSS feeds to view individual comments, the above figures may seriously underestimate the number of visits.
Anyway, it’s clear we have masses of followers. Now it’s time to take over the world.
What if we change the picture like that:
Then we can use following boundary-equations:
Now if we code boundaries of given n-path as and assume linearity then we have
from boundary-equation. The case with forces encodes 2-paths with some “hanging” edges, calibration encodes the way of “binding” these hanging edges (but I don’t know how to explain this in categorical way :)).
Is it ugly analogy? I think it seems like some “duality”.
Zeh, in writing about time, mentions
initial conditions, which may be
analogous to position:
“It has indeed proven appropriate to
divide the formal dynamical description
of Nature into laws and initial
conditions. Wigner (1972), in his Nobel
Prize lecture, called this conceptual
distinction Newton’s greatest discovery,
since it demonstrates that the laws by
themselves are far from determining
Nature.”
I’ll take the risk of suggesting something I not only don’t understand in any depth but also something that may be too specialised to count as concepts rather than specific gadgets, but how about Roger Penrose with:
1. Spinors networks.
2. Twistors.
3. Twistor sheaf cohomology.
Not to be deterred…
How about: charge, current, magnetic flux: Maxwell?
And is there a functorial relationship between those and position, velocity, acceleration?
Of course, anything along these lines was already properly formulated and detailed long ago, but I’m just getting started, so I don’t know anything yet.
How about: charge, current, magnetic flux: Maxwell? And is there a functorial relationship between those and position, velocity, acceleration?
One can formulate all things “charge, current, magnetic flux” along the lines of smooth -functors as in the above example. I talked about that in Charges and twisted -bundles, I.
Doing so unifies phenomena. Though I think it will also show that the triple you mention in not the kind of 1-2-3-sequence analogoues to position-velocity-acceleration, bundle-connection-curvature and category-functor-transformation.
For instance charge and electric current are really two aspects of the same entity, one integral, one local. And magnetic flux is dual.
The best discussion of the subtleties of this after Dirac himself is in the introductory section of Freed’s Dirac Charge Quantization and Generalized Differential Cohomology
Thank you for the comments, Urs. Your Charges and Twisted n-Bundles, I post provides exactly the kind road map needed to guide the thinking and motivate the tools required to understand your approach. This kind of high-level overview is indispensable as a guide to my studies as I assemble a minimal set tools and techniques necessary to understand the mathematics of modern physics. I can not hope to contribute; I can not hope to solve problems; but I can hope to understand how the pieces fit together and basically how each piece works. My dream is someday to see a guide for the intelligent, motivated, undergraduate-level non-specialist that leads one step-by-step, starting with the most basic concepts, through all the relevant fields of math (no proofs required, but no shirking the tough topics either), to the solution of a simple but uniquely contemporary physical problem. John and you and others have already done much work in this direction. Maybe a corner of the n-Lab would be a good place to start gluing all the necessary pieces together. I know you are too busy creating to teach kindergarten classes. But maybe someone out there can help.
How about:
Fourier series
Convergence
Sets
Someone’s been reading Everything and More. But no, even though considering one may lead one to have to consider the next, the original goal was to explain Fourier series, not to explain sets.
My first thoughts were:
1. Paths and loops
2. Homotopies
3. Fundamental groups
(Poincare?)
Just posted and quite relevant
arXiv:0902.0177
Date: Mon, 2 Feb 2009 00:45:24 GMT (49kb)
Title: Operadic bar constructions, cylinder objects, and homotopy morphisms of
algebras over operads
Authors: Benoit Fresse
Categories: math.AT
Comments: 65 pages
MSC-class: 18D50; 18G55
\
The purpose of this paper is twofold. First, we review applications of the
bar duality of operads to the construction of explicit cofibrant replacements
in categories of algebras over an operad. In view toward applications, we check
that the constructions of the bar duality work properly for algebras over
operads in unbounded differential graded modules over a ring.
In a second part, we use the operadic cobar construction to define explicit
cyclinder objects in the category of operads. Then we apply this construction
to prove that certain homotopy morphisms of algebras over operads are
equivalent to left homotopies in the model category of operads.
\ ( http://arxiv.org/abs/0902.0177 , 49kb)
How about examples of questions where the third step is where things go wrong?
For example, in ordinary differential equations? In one dimension everything is “trivial”: everything boils down to monotone regions; in two dimensions, again the asymptotics are generically straight-forward (because separatrices actually separate different dynamics). But in three dimensions, we typically get chaos!
Or how about simplicial cycles? In an n-cycle, the interior of an n-cell is a good manifold, and so is the interior of an (n-1)-cell and even (!) the interior of an (n-2)-cell; but in codimension 3, we can get singular loci. Or perhaps you might consider that something going wrong at the *fourth* step, I don’t know…
Has anyone any less-geometric examples?
My questions about come from my scribblings here:
==================
Formulation
First, due to the symmetry of the diagram above, we can consider the simplified region
This diagram represents a very simple strict 2-category we will denote by with
We will explicitly construct a 2-transport functor
where
===============
I was initially going to write , but thought the simple example could be described by , but then thought you might describe as , but then recognized that as .
As you can tell, I am floundering about, but not giving up. Thanks again for your help.
Eric,
as I said, a 2-functor has to send objects of a 2-category to objects of a 2-category, so has to be an object of a 2-category. But is supposed to denote an algebra, not a 2-category, as far as I understand. This algebra, however, may be regarded itself as being an object of a 2-category. So writing would parse. does not quite parse.
, but then recognized that as .
Let’s see, maybe ou are reading the notation different from its intention: is supposed to denote the Clifford algebra of the vector space equipped with the canonical bilinear form which has entries 1 and entries -1 on its diagonal.
So can be identified with , not . is the real algebra generated by a unit and a single new generator which squares to -1. This is .
But is generated from a unit, a generator that squares to +1 and yet another generator which squares to -1 such that and anticommute.
Hi Urs,
Thanks for your patience.
I am stuck on this idea of making “elements” of “objects” of a 2-category.
Since is an algebra, then if we declare “objects” of our 2-category to be elements of , then “morphisms” are also elements of .
I haven’t worked it out yet, obviously, but this is kind of what I’m thinking…
I want objects of my 2-category to be vectors in intended to represent “position”.
I want morphisms of my 2-category (to be)/(come from) vectors in intended to represent “velocity”. I haven’t settled on how a velocity vector becomes a morphism. I see two possibilities:
or
for some rotor derived from .
I started out thinking about the first option, but then have been coming around to the second. I’m still not sure…
Finally, I want my 2-morphisms to come from acceleration, but I think what I really want to work with is the acceleration bivector . Here again, I’m not sure I want to deal directly with or some rotor derived from .
The slogan I’m after is
Velocity translates position.
Acceleration rotates velocity.
This should fit neatly into the analogy that started me down this path
I don’t think I’m trying to do anything other than what you already said quickly in passing, but there is no way I can understand what you said initially due primarily to language barriers. Everything here is my feeble attempt to translate what you said.
By the way, I borrowed the idea of vectors as objects from Step 3 in
Exercise in Groupoidification.
So maybe what I have in mind is an action 2-groupoid?
Alas, I haven’t been thinking hard enough about this thread to make truly useful comments. So, I’ll make one that’s probably useless.
Eric wrote:
Since is an algebra, then if we declare “objects” of our 2-category to be elements of , then “morphisms” are also elements of .
Why? Is this something you want to be true? You’re making it sound like something that must be true, by the sheer force of pure mathematics. But I only know two ways to make algebras into the objects of a 2-category, and for neither of these are the morphisms given by algebra elements!
Way 1. There’s a strict 2-category where:
Since an algebra is a -enriched category with one object, we can think of the above more elegantly as a full and 2-full sub-2-category of the 2-category of -enriched categories, -enriched functors and -enriched natural transformations. The funny-looking equation above is really the definition of natural transformation in disguise.
Way 2.
There’s a weak 2-category where:
I discussed this second construction in detail in week209. In week211 I explained how it generalized from algebras to superalgebras… and here, interestingly, Clifford algebras play an incredibly important role!
But, I’m not claiming this will be helpful to you. I don’t see how either of these 2-categories is relevant to your current quest.
Thanks! I’ll have to think about the “Two Ways” you outlined. I think I had a different way in mind, which means my way is likely wrong.
Looking back at the Exercise in Groupoidification, I was reminded of something I called
Since an algebra is a monoid in Vect, then
is a category with one object for each vector in the underlying vector space of . Each morphism of “explodes” into a huge spaghetti mess of morphisms in , i.e. for each morphism
in , we get a ton of morphisms
in . One for each vector in .
I was thinking, and could very well be wrong, that if we begin with an algebra and “explode” it so that each vector is an object, then any other object can be thought of as a morphism, i.e. given , then given any other , then
So maybe I am thinking of an action groupoid ?
By the way, if that made any sense (I won’t hold my breath) and elements of are both objects AND morphisms, then I thought they must also be 2-morphisms, and 3-morphisms, etc, etc.
In other words, I thought maybe that exploding an algebra gave you an -category.
Oh oh! I think what I had in mind is related to Way 1, but maybe Way 1.
Exploding a 2-category gives another 2-category in an obvious way. If we consider Way 1, but with only one object , then exploding this gives what I had in mind I think.
If you only have one object, is Way 1 closely related to Way 2?
Re: The Third Time is the Charm
For Alexander Grothendieck the number would be more like ten. At least.