1	N-fold operads: braids, Young diagrams, and dendritic growth. Stefan Forcey, Tennessee State University
2	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.
3	Outline Examples of 2-fold monoidal categories. Operads in 2-fold monoidal categories. Examples of operads. Crystals! Open questions.
4	I. 2-fold monoidal categories.
5	Examples of 2-fold monoidal categories. 1) A braided category V.
6	2) The non-negative ordered integers.
7	3) Young diagrams with lexicographic ordering by columns.
8	…example 3) cont.
9	4) Young diagrams with lexicographic ordering by columns.
10	…example 4) cont.
11	…example 4) cont.
12	II. 2-Fold Operads Recall 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:
13	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:
14	III. Examples of 2-fold operads. 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
15	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.
16	A sequence of Young diagrams C(n) with C(1)=0 is an operad iff the height of the first columns f(n) obeys:
17	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:
18	More examples: minimal operads in Young diagrams 6)
19	7)
20	8)
21	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:
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23	Operad model of “Crystal growth” C_{0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6}vs. RoundDown(55(n-2)/100-Sin(56(n-2)/100)
24	The second observation made by the crystal gazers was:
25	"The rate of growth and..." 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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27	V. Some open questions/projects.
28	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