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.