Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, June 25, 2008

M Theory Lesson 201

It is confusing that we sometimes talk about 0, 1 and 2, and sometimes about 1, 2 and 3, when we really mean the same thing. But one denotes truth values by the former triple and elements of sets by the latter, even if the set contains the number 0. Of course it doesn't matter what one calls objects in a logos, so long as one is careful to explain what structure is being described. Let's stick with 1, 2 and 3 today, because this is conventional notation for the permutations on three objects, given as usual by the $3 \times 3$ circulant matrices with entries 0 and 1.

In logos theory multicategories are more important than ordinary categories, not least because operads are examples of multicategories. Consider the basic triangle category, with only three non-identity arrows. If the triangle is viewed as a multicocategory, what arrows can we draw with it? Any number of inputs is allowed, but for the category 3 repeats soon become inevitable. Heavy use of the identity arrows is made. Now consider a triangle with two way arrows between distinct objects. A pair of two way arrows can represent a 2-cycle permutation on two objects, denoted by the Pauli matrix $\sigma_{x}$. But then naive composition of arrows does not give the composition of 2-cycles in $S_{3}$. To obtain such a 3-cycle it is more natural to involve multiarrows! That is, let a trivalent vertex represent the 3-cycle, as we often do in M Theory.


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