Duality

A duality is a correspondence between two structures where everything in one has a mirror in the other, but with the directions reversed. If you flip all the arrows in one structure, you get the other.

In category theory, every category has an opposite category formed by reversing all morphisms. A statement that is true in a category is also true in the opposite category, but with all arrows reversed. Products become coproducts. Limits become colimits. Monomorphisms become epimorphisms. This is why duality is powerful — every theorem you prove gives you a second theorem for free.

A duality between two concepts means they occupy the same structural position viewed from opposite sides. Supply and demand are dual — one is the buyer’s perspective, the other is the seller’s, and the market is the structure they share. Cost and value are dual — marginal cost is the supply-side boundary of a transaction, willingness to pay is the demand-side boundary, and the price is where they meet. Acquisition and retention are dual — conversion rate measures gaining customers, churn rate measures losing them, and the business is the structure between.

Duality is symmetric. If A is dual to B, then B is dual to A. It is not the same as opposition or contrast — dual things share structure. Contrasting things may have nothing in common except being meaningfully different.

When this library uses dual-of as a frontmatter predicate, it means the subject and object are the same shape seen from opposite directions. Reversing the perspective that defines one gives you the other.