occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Monday, September 01, 2008

M Theory Lesson 220

A simple example of an adjunction between two 1-categories is a Galois connection. We are interested in the case of sets with a partial order, so there exists an arrow from $a$ to $b$ in a set $S$ whenever $a \leq b$. The natural functions (in this case functors) between such sets (here considered as categories) are monotone functions, so a Galois connection consists of two monotone functions $f: S \rightarrow T$ and $g: T \rightarrow S$ such that $f(a) \leq x$ iff $a \leq g(x)$. Observe that, at least when $g$ is onto, $fg$ is idempotent as a map from $T$ to $T$. This follows from $gfg (y) \geq g(y)$ (and $gfg (y) \leq g(y)$), which is true because the application of the one rule to $f(x) \leq f(x)$ gives $x \leq gf (x)$, and we can find a $y$ such that $x = g(y)$.

Usually the sets $S$ and $T$ are quite different. Consider the original example for finite number fields. The finite field with $p^{n}$ elements, for $p$ prime, is an extension of the field with $p$ elements. Take the finite set of all fields in this large field which contain the field with $p$ elements. For any such field $K$, there is a map $K \mapsto \textrm{Gal}(K)$ which sends $K$ to its Galois group in the large field. The dual connection map takes any subgroup $H$ of the main Galois group to all elements of the large field that are fixed by $H$.

Now let $p = 3$ and $n = 3$. There are fields of 3, 9 and 27 elements that extend the field $F_{3}$. The main Galois group is all isomorphisms of the 27 element field that fix the field $F_{3}$. The dual connection map sends the trivial group inside this group to the full 27 element field, the whole Galois group to $F_{3}$ and an intermediary subgroup (bit from trit) to the 9 element field. All this is encoded in an elementary pair of arrows between the 2 kinds of three element set.

CarlBrannen said...

Let me make a pathetic effort to translate this into terms I can understand from the point of view of the stuff I'm doing.

"Partial Order" means the ordering of idempotents (or projection operators) by addition. That is, if p, q, and r are idempotents, and p+q = r, then
p less than or equal to r, and
q less than equal to r.

Strict inequality obtains when p and q are non zero.

"intermediary subgroup (bit from trit)" means the group of idempotents generated by products of the Pauli MUB? This is a group with 9 elements, provided you scale the elements appropriately. (And the scaling needed to make this a group is just where the pi/12 comes from in the neutrino spectrum.)

So how did I do?

September 02, 2008 7:30 AM
Kea said...

Gee, Carl, I still have to work this out myself, but yes, we should put all your idempotents into an adjunction, because mathematicians will like it, but it would be a 3 dimensional version. Note that each idempotent is made from the pieces of an adjunction here. What is nice is that the very idea of an adjunction in more general categories is also about little triangles (units and counits).

September 02, 2008 8:14 AM