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 Stefan Forcey, Tennessee State University

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 Iterated monoidal categories are most famous for modeling loopspaces
via their nerves. There is still an open question about how faithful
this modeling is. An example of a 2fold 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 nfold 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 viceversa.
 Nfold monoidal categories generalize braided and symmetric
categories while retaining precisely enough structure to support
operads. The category of nfold 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 nfold 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 ndimensional Young diagrams to answer the open question of whether
every nfold loop space is represented by the nerve of an iterated
monoidal category.

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 Examples of 2fold monoidal categories.
 Operads in 2fold monoidal categories.
 Examples of operads.
 Crystals!
 Open questions.

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 An operad in the nonnegative integers, where
 is a sequence C(j), j>0, for which
 and for which C(1)=0. The first condition simplifies to

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 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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Operad model of “Crystal growth”
C_{0, 0, 0, 0, 0, 1, 2, 3, 4, 5,
6 }_{ }vs.
RoundDown(55(n2)/100Sin(56(n2)/100)

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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 2fold 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 “sidebranch”
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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 [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
