Formal Definition
We define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms.
Let σ be any statement in this language. We form the dual σop as follows:
- Interchange each occurrence of "source" in σ with "target".
- Interchange the order of composing morphisms. That is, replace each occurrence of with
Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions.
Duality is the observation that σ is true for some category C if and only if σop is true for Cop.
Read more about this topic: Dual (category Theory)
Famous quotes containing the words formal and/or definition:
“True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
—Marcel Proust (1871–1922)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)