Kan Extensions

Abstract

If M is a subset of C, any function t : M→A to a non-empty set A can be extended to all of C in many ways, but there is no canonical or unique way of defining such an extension. However, if M is a subcategory of C, each functor T : M→A has in principle two canonical (or extreme) “extensions” from M to functors L, R : C→A. These extensions are characterized by the universality of appropriate natural transformations; they need not always exist, but when M is small and A is complete and cocomplete they do exist, and can be given as certain limits or as certain ends. These “Kan extensions” are fundamental concepts in category theory. With them we find again that each fundamental concept can be expressed in terms of the others. This chapter begins by expressing adjoints as limits and ends by expressing “everything” as Kan extensions.