# Duality

Any arrow land has an associated arrow land, called its dual, with the same arrows but its composability is the reverse of the original. Since reverse is self-inverse, the dual of the dual is the arrow land with which you started. The dual of an arrow land L is reverse(L).

Throughout the study of arrow lands (and their special cases), duality will enable us to turn statements about arrows in some land into statements about arrows in its dual. For instance, an arrow's Prior in one arrow land is its Post in the dual arrow land, and vice versa: we say, thus, that Prior `is dual to' Post. Consequently, Hom, Parallel (and parallel) are self-dual. The dual of a product of arrow lands is the product of their several duals (and order is preserved).

In general, any property (of an L-arrow) expressed entirely in terms of L implies a dual property, namely the property possessed by any arrow, in L, which possesses the given property in reverse(L): this may be the same as the original, but is often complementary to it. In the fine art of designing notation and choosing which things to define (which I am struggling to learn), there is much to be said for finding a property which is suitably `independent' of its dual, yet simple and producing a subtly more powerful property when combined with this dual.

It is often interesting to take a property expressed in the terms of an arrow land and replace some condition in its definition with its dual. The resulting property can be highly interesting: when combined with the original, it often produces a compound property of great power and simplicity. It can, equally, be highly instructive to take a familiar property (especially a self-dual one) and express it as such a compound. The art lies in choosing a base property and defining it in terms which include a condition which, when combined with its dual, produces the right results. It is often best to ensure that this condition is, as far as possible, independent of the other conditions in the definition: this gives a form of conceptual orthogonality to the dual pair.

When I identify a property which appears to be reasonably independent of its context, I take a look at its dual. If that suggests a more plausibly independent notion, I'll pass that notion back through the duality and compare it to the original. I find this instructive.

As with duality, so also with any other self-inverse re-expression of any part of a theory. One can look for conditions which decompose into pairs of conditions, one the re-expressed form of the other: this can simplify the expression of quite complex properties and often helps clarify proofs. For some such re-expression, suppose we have a property which we think of as a left-form of a property which the re-expression converts to its right-form: ie think of the re-expression as chiral. The property's dual will usually also be describable in terms of a left-form and a right-form (but beware: the language which comes more naturally to describing the property and its dual may leave us with the left-form of one property dual to the right-form of the other - duality is often chirally active). In such cases, the left-form and its dual may combine interestingly: the left- and right-forms of the property may combine interestingly, as may those of the dual property; and all four forms together can be powerful despite great simplicity in the basic property. You can find an example of this in my definition of the Group.

Written by Eddy.
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