N-fold operads: braids, Young diagrams, and dendritic growth.

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N-fold operads braids, Young diagrams, and
dendritic growth.

• Stefan Forcey, Tennessee State University

F,D,W,K
1)
2)
F,S,S
What do these two sequences of pictures have in
common?
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Abstract

• Iterated monoidal categories are most famous for
  modeling loop-spaces via their nerves. There is
  still an open question about how faithful this
  modeling is. An example of a 2-fold monoidal
  category is a braided category together with a
  four strand braid from a double coset of the
  braid group which will play the role of
  interchanger.
•  Examples of n-fold monoidal categories include
  ordered sets with n different binary operations.
  For each pair of operations an inequality
  expresses the interchange. We will present
  several example sets with their pairs of
  operations, beginning with max and plus on the
  natural numbers and proceeding to two new ways of
  adding and multiplying Young diagrams. The
  additions are vertical and horizontal stacking,
  and the multiplications are two ways of packing
  one Young diagram into another based respectively
  on stacking first horizontally and then
  vertically, and vice-versa.
•  N-fold monoidal categories generalize braided
  and symmetric categories while retaining
  precisely enough structure to support operads.
  The category of n-fold operads inherits the
  iterated monoidal structure.  We will look at
  sequences that are minimal operads in the totally
  ordered categories just introduced, and discuss
  how these sequences grow. It turns out that the
  later terms are completely determined by the
  choice of initial terms, and if this choice is
  made carefully there appears a remarkable
  correspondence to certain natural processes. In
  fact, the growth rate of physical dendrites such
  as metallic crystals and snowflakes oscillates in
  a way directly comparable to that of our operads.
  
•  In relation to other topics at the conference,
  we will pose some open questions about how the
  nerves of n-fold operads might be described, and
  whether or not they do indeed form dendroidal
  sets. If time permits we will also discuss the
  possibility of using our families of
  n-dimensional Young diagrams to answer the open
  question of whether every n-fold loop space is
  represented by the nerve of an iterated monoidal
  category.

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Outline

1. Examples of 2-fold monoidal categories.
2. Operads in 2-fold monoidal categories.
3. Examples of operads.
4. Crystals!
5. Open questions.

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I.2-fold monoidal categories.
The important things to remember about a 2-fold
monoidal category 1) It has two tensor
products 2) The second product is a
monoidal functor with respect to the first
3) For coherence and unit axioms see B,F,S,V.
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Examples of 2-fold monoidal categories.

• 1) A braided category V.

or h
h s2
In fact, h can have any underlying braid which
lies in both double cosets Hs2K and Ks2H for
subgroups H and K of B4 as described in F,H
(s2-1 works also).
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2) The non-negative ordered integers.
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3) Young diagrams with lexicographic ordering by
columns.
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example 3) cont.
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4) Young diagrams with lexicographic ordering by
columns.

The (ambiguous) packing is
(we use this step for both tensor products
and )
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example 4) cont.

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example 4) cont.

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II. 2-Fold OperadsRecall that an operad in a
braided category is a sequence of objects
C(n)with a composition where j is the sum of
the j i. The composition obeys this sort of
commuting diagram, where an element of C(n) is
represented by an n-leaved corolla

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The important thing to notice here is that the
interchange h is all we need in order to sensibly
demand the associativity axiom. The shuffle is
replaced by the interchanger in the following
manner

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III. Examples of 2-fold operads.

• An operad in the non-negative integers, where
• is a sequence C(j), jgt0, for which
• and for which C(1)0. The first condition
  simplifies to

1) Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13,
21, .
2) Counting numbers 0, 1, 2, 3, .
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A minimal operad in the non-negative integers
relative to a given string 0, C(2), , C(k) is
defined by extending the sequence C(n)
maxC(i) C(n-i)i1n-1 for n gt k.
(minimal in the sense that each term is no larger
than needed.)
3) C0, 1 0, 1, 1, 2, 2, 3, 3,
4) C0, 1 ,1, 2, 3 0, 1, 1, 2, 3, 3, 4, 4, 5, 6,

5) C0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 60, 0, 0, 0,
0, 1, 2, 3, 4, 5, 6,
6, 6, 6, 6, 6, 7, 8, 9, 10, 11, 12,
12, 12, 12, 12, 12,
13, 14, 15, 16, 17,
OPEN question find a closed formula for these
sequences. NOTE The terms oscillate around
linear growth.
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A sequence of Young diagrams C(n) with C(1)0
is an operad iff the height of the first columns
f(n) obeys

This means that the first column height will form
an operad in the max, category.
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A minimal operad in Young diagrams relative to a
given string 0, C(2), , C(k) is defined by
extending the sequence for n gt k using
lexicographic max

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More examples minimal operads in Young diagrams

• 6)

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7)

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8)
OPEN Question find a closed formula for these
sequences of diagrams. NOTE The first column and
the remaining structure take turns growing.
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IV. Crystals

• In
• Growth pulsations in symmetric dendritic
  crystallization in thin polymer blend films
• Vincent Ferreiro, Jack F. Douglas, James Warren,
  and Alamgir Karim describe the measurements they
  took in 2002 as certain crystals formed in
  solution. Their first observation was

F,D,W,K
1) The growth of the radius oscillates around
linear growth.
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F,D,W,K
Delta R is the radius length less the line around
which it oscillates. (slope average growth
rate) The change in radius is basically
sinusoidal. Experimentally, amplitude depends on
temperature and period depends on film thickness
of the solution.
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Operad model of Crystal growth C0, 0, 0, 0,
0, 1, 2, 3, 4, 5, 6 vs. RoundDown(55(n-2)/100
-Sin(56(n-2)/100)
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The second observation made by the crystal gazers
was
F,D,W,K
2) The radius and the width of an arm of the
crystal take turns growing.
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• The rate of growth and period of oscillation of
  the width of the dendritic crystal arm appear to
  be roughly the same as the corresponding numbers
  for the radius of the dendrite, but precisely out
  of phase.
• Recall that the terms of 2-fold operads of Young
  diagrams, minimal relative to a given starting
  sequence, do take turns increasing in size
  vertically and horizontally. However, the length
  of a side-branch grows only logarithmically.
• But, the number of blocks in the entire structure
  grows linearly. Thus the number of blocks to the
  right of the first column grows at the same
  average rate as the height of the first column.
  These two growth rates oscillate out of phase,
  just as is seen in the crystal.

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The rate of growth and period of oscillation of
the width of the dendritic crystal arm appear to
be roughly the same as the corresponding numbers
for the radius of the dendrite. This seems to
be related to the self similarity of the
crystal. The authors describe multiple possible
physical mechanisms for the out of phase growth
pulsations. They theorize in general that the
pulsations are strongly connected to the geometry
of the crystal, and therefore probably arise in
other side-branching crystals, such as
snowflakes. Much more experimentation with the
operad model needs to be done before any
implications are suggested.

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V. Some open questions/projects.

• Find non-recursive formulas for the general
  minimal operads.
• Investigate operads of Young diagrams with the
  packing multiplication. These grow
  exponentially.
• Describe the nerves of n-fold operads.
• Find new 2-fold monoidal structures which
  support operads that are even better at modeling
  crystal growth. How about other sorts of
  (self-similar) growth? Organisms, speleothems,
  networks, biological colonies, etc. In general
  Allometric measurements.
• Use non-commutative versions of the category of
  Young diagrams to model loop spaces.

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References
B,F,S,V C. Balteanu, Z. Fiedorowicz, R.
Schwanzl, R. Vogt, Iterated Monoidal
Categories, Adv. Math. 176 (2003), 277-349.

• F,D,W,K
• Vincent Ferreiro, Jack F. Douglas, James Warren,
  Alamgir Karim
• Growth pulsations in symmetric dendritic
  crystallization in thin polymer blend films
• PHYSICAL REVIEW E, VOLUME 65, 051606

F,S,S STEFAN FORCEY, JACOB SIEHLER and E. SETH
SOWERS OPERADS IN ITERATED MONOIDAL
CATEGORIES Journal of Homotopy and Related
Structures, vol. 2(1), 2007, pp.143 F,H Stefan
Forcey, Felita Humes, Equivalence of
associative structures over a braiding. To
appear in Algebraic and Geometric Topology.