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Stroop Theory (lives in category land)

  1. Jun 1, 2006 #1

    marcus

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    if there is any place where string ideas and loop quantum gravity might join hands and live happily ever after it is in the Baez paper
    quant-ph/0404040
    and a few other papers to which this one is the gentlest introduction I have found so far.

    this is where Baez introduces the idea of a "star" category
    and more specifically a TENSOR star category-----(a star category with something like a tensor product defined------that is not the official math term for it but for brevity sake that is what I will call this type of category).

    the category of SPACETIMES is a tensor star category

    the category of HILBERTSPACES is a tensor star category

    This could be a giveaway hint that quantum gravity might eventually INVOLVE THIS sort of CATEGORY---because QG is about connecting spacetime dynamics to a hilbertspace of quantum states. This type of category is rather special and it is a remarkable coincidence that hilbertspaces (which is where quantum states are vectors in)
    and spacetimes (which is what Gen Rel is the dynamics of) should both be this same type of dingus.
    =================

    now I have already made several faux pas of math language, so I have to back up. But the message i want to get across, before doing that, is that we should find out what this kind of category is.

    It will turn out that string WORLDSHEETS and non-string QG SPINFOAMS come up rather naturally in this context
     
    Last edited: Jun 2, 2006
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  3. Jun 1, 2006 #2

    marcus

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    The hard part is the categorical definition of tensor product. The STAR part is easy.
    If you look into quant-ph/0404040
    you will quickly see what a "star" category is. that is no problem.
    It is just a category where every map f:X->Y has a DUAL map f*:Y->X
    which is its PARTNER
    (CATEGORISTS CALL MAPS "MORPHISMS")
    and a simple example to consider is Hilbertspaces with the morphisms being continuuous linear maps----think matrices----and the dual of a map being the familiar ADJOINT map----think about flip-conjugating the matrix. We do this as freshmen. All the categorists are doing is ABSTRACTING the idea of adjoint map.

    and it is a nice reciprocal thing: f**=f, and (fg)* = g*f* and the IDENTITY map is its own adjoint. Just as you expect doing it with matrices in the example
    =========================

    "star" category is Baez made-up terminology which he says might disturb some serious Categorists. Some of them like to always use the correct terminology and they might say "A Category with Dual Morphisms". they are stubborn sum*****es and always want to force you to use their nomenclature.

    But in this thread we follow Baez 0404040 and call it "star" because the star reminds us of the adjoint symbol f*
    =========================

    Categorists would probably call a TENSOR star category by the longer correct term
    "symmetric braided monoidal category with dual"

    but for our purposes, a "symmetric braided monoidal" is too long to say and that kind of category is simply one that has something like a TENSOR PRODUCT defined on it.
    =========================

    the absolute minimum you need to understand what is going on is to check out an undergrad algebra text and be sure you understand what the tensor product of two hilbertspaces is.

    there are a couple of thin books by Paul Halmos that used to be good for this. but by now there must be a ton of other. it takes a couple of pages.
    =========================

    One always suspects, after first being confronted with a generous helping of indigestible terminology, that mathematicians (especially Categorists) are crazy.

    this is a natural and i believe basically reasonable reaction.

    However it is uncalled for in this case because you have not seen the PICTURES yet.

    So best you go to Baez 0404040 and look at the pictures and you will see pictures of string worldsheets and QG spinfoams and you will see a PICTORIAL ANALOGY TO FEYNMAN DIAGRAMS and you will see that these things are recognizable as spacetimes, sort of hoses that connect a "before" space to an "after" space. These spacetime hoses that connect before to after are a concrete manifestation of a dynamic PROCESS and the traditional mathematician's name for them is COBORDISMS.

    And these cobordisms (which are spacetiime dynamic process in a very pictorial form) ALSO FORM A TENSOR STAR CATEGORY.

    The pictures are very simple, almost self explanatory. Putting two cobordisms (call them worldsheets or call them hose diagrams, whatever) side by side make their TENSOR PRODUCT and putting one on top of another and welding makes the composition of mappings. And the DUAL is just flipping upside down so you get the process running in reverse----exchange "before" with "after".

    this seems so familiar as to be deja vu. We may have had a PF thread about this in April 2004 when the paper came out. It was the run-up to Rovellis Loop/Spinfoam conference at Marseille.
    ===================

    so all I can say is probably some of us should review quant-ph/0404040 and a couple of related papers like the Baez Chronology of Physics Since Maxwell (which is physics from a categories viewpoint)

    If you know something about categories you know that if you have two categories and you find a way to ANALOGIZE between them-----like a way to connect relativityspaces to hilbertspaces----the analogy is called a FUNCTOR.

    ultimately a Quantum Gravity, or a General Relativistic quantum physics might appear to some frazzled unshaven guy on a mountain as a FUNCTOR from something like Cob to something like Hilb-----from the spacetime and matter processes that connect before to after-----to the hilbertspace of quantum states that represents a persons knowledge about the system. So he will come running down from the mountain yelling "I HAVE SEEN THE GREAT FUNCTOR"

    there is a real danger of this happening, so we probably all ought to know how to say Wie geht's and Merci beaucoup in Category language.
     
    Last edited: Jun 2, 2006
  4. Jun 1, 2006 #3

    marcus

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    this is just to correct terminology
    Baez called the first one of these 4Cob (Cob is short for cobordism)
    and it is the category where the objects are 3D spaces
    and the morphisms are 4D spacetimes that connect a "before" 3D space to an "after" one----IOW "4-cobordisms"

    Baez called the second one Hilb
    and it is the category where the objects are hilbertspaces and the morphisms that connect them are bounded linear operators, or continuous linear maps, whatever you want to call them: matrices basically.

    because Hilb is a STAR category, and every morphism has a dual morphism, there is the idea of a SELF ADJOINT map, where f*=f
    and those are the OBSERVABLES in a quantum theory.
    =====================

    BTW I should post a warning (which I think Baez makes several places, if not this paper some other closely related paper).
    the warning is that THIS MIGHT ALL NOT WORK. it might be useless.

    a lot of beautiful elegant mathematics turns out not to solve the problem---not do what you want it to, in the end.

    For us at PF, these tensor star categories have been around since spring of 2004, and for some people here, like Kea, a lot longer.

    I am only just now beginning to think I have to buckle down and learn more about it. Of course it might not work out. But recent stuff connected with Baez 31 May Colloquium at Perimeter is making me think it is too risky not to know the tensor category basics.
    Ordinary category basics I sort of know, tensor ("symmetric braided monoidal") category basics I dont know. Maybe some others are in the same boat.
     
    Last edited: Jun 2, 2006
  5. Jun 2, 2006 #4

    marcus

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    Last edited: Jun 2, 2006
  6. Jun 2, 2006 #5

    arivero

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    Lets try Kea, I am unable to help here.

    In fact I am amazed about how category theory enters play. A lot of people expected it to have arole in physics via morphism between representations in Hilbert space; Doplicher did some work on it with the so-called intertwinning operators, and this is related to Morita equivalence. In some sense it was the low-profile role of categories. A "high-profile role" was expected to come via topoi theory, or perhaps via sheaves. It is amusing that Baez approach does not fit with these a-priori expectations, but follows an original path.
     
  7. Jun 2, 2006 #6

    Hurkyl

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    Hrm. How about a T* category? :smile: We already have B* and C* algebras, and H* categories!


    But you're not supposed to! :smile: Part of the whole point is that this is higher-dimensional algebra, and requires higher-dimensional formulas! So instead of our boring one-dimensional equations, we're supposed to get used to two-dimensional equations. (No, I'm not used to two-dimensional equations, but I'm getting there) And there's this whole conjecture that higher category theory will just turn out to be topology in disguise, which is good, because category theory is algebra!



    So we have this funky diagram:

    Code (Text):
                   |
                   |
        /-\        |
       /   \       |
      /     \      |
      |      \     /
      |       \   /
      |        \-/
      |
      |
     
    What does it mean?!?! Well, for simplicity, let's assume this curve is oriented downwards, and that we have a canonical element f of our T*-category. So, everywhere this curve is heading downwards denotes f, and upwards denotes f*.

    But wait! There are two secret lines in this diagram... it really should look like:

    Code (Text):

         :         |
         :         |
         ^         |
        / \        |
       /   \       |
      /     \      |
      |      \     /
      |       \   /
      |        \ /
      |         v
      |         :
      |         :
     
    where the dotted lines denote the multiplicative identity.

    But wait, there are more secrets! The diagram is hiding some important dots:

    Code (Text):

         :         |
         :         |
         O         O
        / \        |
       /   \       |
      /     \      |
      |      \     /
      |       \   /
      |        \ /
      O         O
      |         :
      |         :
     
    So what are these dots? They represent the morphisms in our T*-category. The upper-left dot is our counit map, and the lower-right dot is our unit map. The other two dots are identity maps.


    So now how do we read it? Well, we have to break it into pieces:

    Code (Text):

         :         |
         :         |
         :         |
         :         |


         :         |
         O         O
        / \        |


       /   \       |
      /     \      |
      |      \     /
      |       \   /


      |        \ /
      O         O
      |         :


      |         :
      |         :
      |         :
      |         :
     
    The first, third, and fifth parts of this picture denote objects in our T*-category. The second and fourth parts denote morphisms in our T*-category. This whole picture is simply a diagram denoting the composition of two morphisms! We simply drew our objects as big pictures, and our morphisms as filaments between the pictures!



    Code (Text):

         :         |
         :         |
         :         |
         :         |
     
    I've already mentioned that the dotted line denotes the identity element 1, and the dashed line denotes our canonical element f. Well, placing things side-by-side denotes tensor products. This is nothing more than the object [itex]1 \otimes f[/itex].


    Code (Text):

         :         |
         O         O
        / \        |
     
    The dot on the left denotes the counit map i, and the dot on the right denotes the identity map 1. (Sorry I use 1 so much) So, this is simply the morphism [itex]i \otimes 1[/itex]. This morphism points from the object above to the object below.

    Code (Text):

       /   \       |
      /     \      |
      |      \     /
      |       \   /
     
    This is the object [itex]f \otimes f^* \otimes f[/itex]

    Code (Text):

      |        \ /
      O         O
      |         :
     
    The one on the right, remember, is the unit map e. So, this is simply the morphism [itex]1 \otimes e[/itex].


    Code (Text):

      |         :
      |         :
      |         :
      |         :
     
    And this is simply the object [itex]f \otimes 1[/itex].



    If we were being traditional, this diagram would be drawn as:

    Code (Text):

             i o 1                 1 o e
    1 o f  -------->  f o f* o f --------> f o 1
     
    So, this denotes nothing more than the product:

    [tex](i \otimes 1) \cdot (1 \otimes e)[/tex]


    Since we have a canonical way to replace [itex]f \otimes 1[/itex] and [itex]1 \otimes f[/itex] with f, we can view the above as a map f ---> f, and the zig-zag identity simply asserts that this is equal to the identity!


    I guess if we wanted to be complete, we could explicitly add in the isomorphisms that map [itex]f \rightarrow 1 \otimes f[/itex] and [itex]f \otimes 1 \rightarrow f[/itex]:

    Code (Text):

              |f
              |
              |
              o
             : \
            :   \f
         1 :     \
          :       \
         :         |
         :         |
         :         |
         O         |
        / \  *     |
       /   \f      |
    f /     \      |f
      |      \     /
      |       \   /
      |        \ /
      |         O
      |         :
      |         :
      |         :
      \         :1
       \       :
        \     :
         \   :
          \ :
           o
           |
           |
           |f
     
     
    Last edited: Jun 2, 2006
  8. Jun 2, 2006 #7

    selfAdjoint

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    I was interested to see that the "star" operation is a pre-sheaf. In the most general definition (see Spanier's Algebraic Topology) a pre-sheaf is just a contravariant functor, say F, that takes objects and morphisms A -> B into F(B) -> F(A).

    Marcus, you say we need to get our minds around tensor product of Hilbert spaces, and that's true, but I also think that monoidal is a block to understanding. Baez gives a formal definition but we need to noodle around with it to see how it - and particularly that commutative diagram - really work. Also the non-cartesian property. Only when we are very clear iin our minds what these "stroop" categories do and are can we move forward.
     
    Last edited: Jun 2, 2006
  9. Jun 2, 2006 #8

    marcus

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    Thanks Alejandro, selfAdjoint, and Hurkyl.
    Hurkyl I hope writing this helpful post didn't make you seriously late for work!
     
  10. Jun 2, 2006 #9
    I've also been thinking about how strings might unite with LQG, but it does not have directly to do with category theory. Is this thread open to discussion without category theory?
     
  11. Jun 2, 2006 #10

    marcus

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    Mike regretfully I think not
    I would like this thread to help as a CRUTCH to understanding these papers which I mentioned before:

    the April 7 2004 version of Baez 0404040 is here
    http://math.ucr.edu/home/baez/quantum/
    the May 31 perimeter colloquium page is here
    http://math.ucr.edu/home/baez/quantum_spacetime/
    the actual slides/lecturenotes for the colloquium talk are here
    http://math.ucr.edu/home/baez/quantum_spacetime/qs.pdf

    this is not just categority in GENERAL but actually a particular AREA of applied category theory
    which is the inscrutable and unexpected application of category theory to topological quantum field theory and ultimately quantum gravity.


    If we want to make headway on the main thing, then I dont think we can efficiently go outside category theory here, and should try to gnaw away at what Baez calles the "HIGHER ALGEBRA" approach to Gen Rel and QM.
     
    Last edited: Jun 2, 2006
  12. Jun 2, 2006 #11

    Hurkyl

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    I've rewritten my earlier post. It's better now! :smile: (Though I don't mention the connection between this and 2-categories) I've saved off the old text if you still want to see that
     
  13. Jun 2, 2006 #12

    marcus

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    Hear, O Israel: the LORD our God is one:

    5 and thou shalt love the LORD thy God with all thine heart, and with all thy soul, and with all thy might.

    6 And these words, which I command thee this day, shall be upon thine heart:

    7 and thou shalt teach them diligently unto thy children, and shalt talk of them when thou sittest in thine house, and when thou walkest by the way, and when thou liest down, and when thou risest up.

    8 And thou shalt bind them for a sign upon thine hand, and they shall be for frontlets between thine eyes.

    9 And thou shalt write them upon the door posts of thy house, and upon thy gates.

    this is the Schema
    http://www.bible-researcher.com/shema.html
    also known as Deuteronomy 6 verses 4-9
     
  14. Jun 2, 2006 #13

    Hurkyl

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    That's not quite the old text I meant! :smile:
     
  15. Jun 2, 2006 #14

    marcus

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    ay Hurkyl,

    now let's talk about the "creation and annihilation" morphisms that you mentioned, i for initiate and e for erase.

    in category Cob (the category made of spacetimes) the UNIT is just the null manifold, the nada.

    because the tensor mult is simply to put one space BESIDE another.

    so the iH morphism just CREATES TWO UNIVERSES WITH OPPOSITE ORIENTATION, side by side, out of the null so it is a morphism like this:

    iH : nothing -> H x H*

    but to be kosher the x should be a tensorproduct and it is sometimes just not worth going into TEX so I will use the "cool" for tensor and I will say UNIT for "nothing"

    iH : UNIT -> H :cool: H*

    and the PICTURE for iH is simply a fat inverted letter U, like made of tube or spaghetti, a fat intersection sign (if you know set theory)

    ========
    OK that is in the case of category Cob. It is an operation or morphism which creates two equal universes of opposite orientation out of the nada---the unit universe. and of course there is a SPACETIME COBORDISM DRAPED BETWEEN THEM which is the fat inverted U.
    =========

    Now what about HILBERSPACES. what about the category Hilb. There the UNIT is just the complex numbers C, which is the simplest hilbertspace you can have.

    Now here, notice that for a given hilber H, the tensorproduct H:cool: H* is simply some maps from H to ITSELF. You can figure out what those maps are by thinking about multiplying a column vector times a row vector-----i.e. multiplying two n-dim vectors in the "wrong" order so you get an nxn matrix instead of just a number. so H:cool: H* is just some linear maps from H to itself. Now how do we interpret the "creation morphism"?

    iH : C -> H :cool: H*

    well what this does is it takes an complex number z in C, and it sends it to the MAP from H to itself that is simply multiplying a vector by the scalar z! And that is a kind of map that is in H:cool: H* so everything is cool.
    ==============

    Both these categories Cob and Hilb are important so we have to do examples in both cases. now we have done the iH in both cases and still have to do the eH morphism in both cases.

    this is the destructo morphism that annihilates the tensor product H*:cool: H, of the dual of any object H in the category, tensor with itself. This is the ERASURE morphism eH

    In the Cob case it has to give NOTHING. it has to start with two equal universes (with opposite orientation) side by side and it has to conver that to the unit of the category which is the Null manifold where there isnt anything there.

    I think we have enough momentum to see how that goes so I will stop here for a while.
     
    Last edited: Jun 2, 2006
  16. Jun 3, 2006 #15

    arivero

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    By the way, it is interesting to do not become obsessed by the arrow in the cathegorial notation. Axiomatically there is a function "source(a)" and a function "target(a)" ranging for all arrows "a", and that is all. Most cathegories can be dualised by using the functions in a reverse way ("reversing the arrows").
     
  17. Jun 3, 2006 #16

    Hurkyl

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    The matrix "algebra" of all matrices over a field provides a very nice example of the category... and one that people actually use as a category (though they usually don't realize it!). That's a good example of something you usually don't picture with arrows.



    Maybe I'm getting too far ahead, but one of the things that intrigued me is that you can start adding more levels -- instead of considering 3-spaces, and 4-spacetimes between them... you can consider 2-dimensional stuff, 3-space between them, and 4-spacetimes between the 3-spaces. This gets mapped not into Hilb, but 2-Hilb. I spent a bit of time trying to fathom what that all means, but didn't get far enough to make me happy. :frown:
     
  18. Jun 3, 2006 #17

    marcus

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    there is no schedule that you can rush or get out of synch with. this thread is a good place to help us try to fathom the multilevel stuff.

    the big motive (for me) is I see people like Freidel making progress with 3D gravity
    and casting about for ways to extend to 4D

    and one gets the suspicion that they, in desperation, may have to "jack the algebra up" in order to get the nice Freidel things to happen in one higher Dee.

    wilson loops might have to become "wilson homotopies" that are paths between loops------and those might transmute into the faces of a spinfoam---or some such pocus hocus.

    in james thurber's book THE WHITE DEER there is a moment at which "the King gave a great haarooof! like a lion tormented by magical mice" and I sympathize with him, these suspicions that higher algebra may hold keys to extending what already works so well up one Dee, these suspicions itch in my fur and are a bother.

    so please pursue it on any path and schedule you know how!
     
  19. Jun 3, 2006 #18

    Hurkyl

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    I was trying to imagine the next level as sort of like Loop Quantum Gravity. Remember that in a spin network, the nodes represent chunks of space-time, and edges represent the faces between them. The edges are labelled with representations of SU(2), and the vertices with "intertwining operators" between the representations.


    As a warning, this is very much speculation. :biggrin:


    So I was trying to picture the next level of a TQFT in this way. When we're up a level 4Cob consists of:

    0-cells that are 2-dimensional manifolds.
    1-cells that are 3-dimensional cobordisms between 0-cells.
    2-cells that are 4-dimensional cobordism-like things between the 1-cells

    So, I imagined that maybe the 0-cells are our primitive area elements of space -- they are what the edges represent in a spin network. The 1-cells are our primitive volume elements of space -- they are what the nodes of a spin network represent.


    And then on the algebraic side, some objects of 2Hilb look like categories of group representations! Let's say we limit ourself to these, so that our TQFT is a (2-)functor from 4Cob into the 2-category of categories group representations.

    So, to each of our area elements, our TQFT assigns a "state space" that is none other than a category of group representations. In other words, our TQFT assigns a group to each area element. Then, to select a state, we merely pick a representation of our group.


    Then, we have our volume elements which tell us how to get from one product area elements to another product of area elements. In the algebra world, this is a functor that tells us how to transport a group representation on the source side over to become a group representation on the target side... and then when we take the 2-Hilb "inner product", the result is the Hilbert space of intertwining operators.

    So, to construct a state, we pick one of these intertwining operators.


    That's as far as I got -- I couldn't picture how the 4-dimensional stuff works! But that's probably because I never got that far in the LQG picture either. :smile:
     
  20. Jun 3, 2006 #19
    As I understand the situation, recently a guage theory has be shown to be a representation of a string theoy (what paper is that). And also guage theory is connected to knots as talked about in J.C. Baez' book "Gauge Fields, Knots and Gravity". So I have to wonder if string theory might be an alternative description of links and knots? Perhaps the link invariants might equate to the string states, etc.
     
  21. Jun 3, 2006 #20

    marcus

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    I think you both probably realize where I am is page 19 of the "Quantum Spacetime" lecture notes
    http://math.ucr.edu/home/baez/quantum_spacetime/qs.pdf
    From what Hurkyl says, it sounds like he is also on that page. Maybe Mike as well. But just to be extra explict about where I am hung up and what I am looking at I will try to paste it here as well.
    Nope, it doesnt copy. So I will have to play the scribe and trans-type it

    ===quote page 19 and 20===
    In 3D quantum gravity---more generally in any extended topological quantum field theory---we have
    a 2-category where:
    * objects describe kinds of matter
    * morphisms describe choices of space
    * 2-morphisms describe choices of spacetime

    In 3D quantum gravity this matter consists of point particles---see the work of Freidel et al...

    In 4D topological gravity this matter consists of strings---see my papers with Crans, Wise, and Perez.

    3. In higher gauge theory we have fields describing parallel transport not just for point particles moving
    along paths...
    but also for strings tracing out surfaces...
    I've developed this in papers with Bartels, Crans, Lauda, Schreiber and Stevenson...
    ===end of sample===
    [I am leaving out a lot of pictures as i transcribe]
     
    Last edited: Jun 3, 2006
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