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

Stefan Forcey, Tennessee State University

[F,D,W,K]

[F,S,S]

What do these two sequences of pictures have in common?

1)

2)

AbstractIterated 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.

Outline

I. Examples of 2-fold monoidal categories.

II. Operads in 2-fold monoidal categories.

III. Examples of operads.IV. Crystals!V. Open questions.

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 withrespect to the first:

21,

)()()()(: 121212 DBCADCBA

3) For coherence and unit axioms see [B,F,S,V].

Examples of 2-fold monoidal categories.

1) A braided category V. 21

or =

In fact, can have any underlying braid which lies in both double cosets H2K and K2H for subgroups H and K of B4 as described in [F,H] (2

-1 works also).

2) The non-negative ordered integers.

21 ,max

},max{},max{)}(),max{(: DBCADCBA

3) Young diagrams with lexicographic ordering by columns.

stackingverticalstackinghorizontal 21 ,

…example 3) cont.

)()()()(: 121212 DBCADCBA

4) Young diagrams with lexicographic

ordering by columns.

packingverticalpackinghorizontal 21 ,

The (ambiguous) packing is:

(we use this step for both tensor products and )

…example 4) cont.

=

…example 4) cont.

=

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:

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

•An operad in the non-negative integers, where

is a sequence C(j), j>0, for which

and for which C(1)=0. The first condition simplifies to

III. Examples of 2-fold operads.

21 ,max

1) Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, ….

2) Counting numbers: 0, 1, 2, 3, … .

• 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) = max{C(i) + C(n-i)}i=1…n-1 for n > 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, 6=0, 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.

• 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.

• A minimal operad in Young diagrams relative to a given string 0, C(2), …, C(k) is defined by extending the sequence for n > k using lexicographic max:

))}(...)(()({)( 2211...

max1

injj

jCjCiCnCi

6)

More examples: minimal operads in Young diagrams

…

7)

8)

OPEN Question: find a closed formula for these sequences of diagrams.NOTE: The first column and the remaining structure “take turns” growing.

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:

IV. Crystals

1) The growth of the radius oscillates around linear growth.

[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.

[F,D,W,K]

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)

0

10

20

30

40

50

60

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

crystal

operad

The second observation made by the crystal gazers was:

2) The radius and the width of an arm of the crystal “take turns” growing.

[F,D,W,K]

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.

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.

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.

References

[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 CATEGORIESJournal of Homotopy and Related Structures, vol. 2(1), 2007, pp.1–43

[F,H]Stefan Forcey, Felita Humes,

Equivalence of associative structures over a braiding. To appear in Algebraic and Geometric Topology.

[B,F,S,V] C. Balteanu, Z. Fiedorowicz, R. Schw¨anzl, R. Vogt,

Iterated Monoidal Categories, Adv. Math. 176 (2003), 277-349.