Constructions on Categories
Categorical duality is the process “Reverse all arrows”. An exact description of this process will be made on an axiomatic basis in this section and on a set-theoretical basis in the next section. Hence for this section a category will not be described by sets (of objects and of arrows) and functions (domain, codomain, composition) but by axioms as in § I.1.
KeywordsFunctor Category Product Category Natural Transformation Covariant Functor Dual Statement
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- The leading idea of this chapter is to make the simple notion of a functor apply to complex cases by defining suitable complex categories — the opposite category for contravariant functors, the product category for bifunctors, the functor category really as an adjoint to the product, and the comma category to reduce universal arrows to initial objects. The importance of the use of functor categories (sometimes called “categories of diagrams”) was emphasized by Grothendieck  and Freyd . The notion of a comma category, often used in special cases, was introduced in full generality in Lawvere’s (unpublished) thesis , in order to give a set-free description of adjoint functors. For a time it was a sort of secret tool in the arsenal of knowledgeable experts.Google Scholar
- Duality has a long history. The duality between point and line in geometry, especially projective geometry, led to a sharp description of axiomatic duality in the monumental treatise by Veblen-Young on projective geometry. The explicit description of duality by opposite categories is often preferable, as in the Pontrjagin duality which appears (§ IV.3) as an equivalence between categories, or as an equivalence between a category and an opposite category (see Negrepontis ).Google Scholar