Kan Extensions and Nonsensical Generalizations

The categorical concept of Kan extensions form a more general notion of both limits and adjoints. The general definition of Kan extensions is given and motivated by several concrete examples. After providing the necessary background on some basic categorical objects and theorems, the relationship between Kan extensions, limits, and adjoints is expressed through two theorems from [3]. 1. Some Preliminary Categorical Concepts A tremendous array of fields within mathematics draws heavily upon the ideas of limits and adjoints. While these notions are sufficiently general for most uses, there exists a more abstract concept introduced by Kan [2], which encompasses both limits and adjoints. Although the following discussion assumes familiarity with the basic language of category theory, we begin by summarizing some terminology and a few results for reference and clarity. The notion of a limit is the first of these. Limits are easily understood through the auxiliary notion of a cone. Definition 1.1. Given a functor F : D → C, a cone on F is a pair (C, pD) consisting of: • an object C ∈ C, • a morphism pD : C → FD in C, for every object D ∈ D, such that for every morphism d : D → D′ in D, pD′ = Fd ◦ pD. The name “cone” is used for a reason; pictorially, cones are situations in which there are morphisms that take the object C to the objects FDi, with the following diagram commuting: C pD1 {{ww ww ww ww w pD2 (( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q FD1 F (d) // FD2 ... FDi Definition 1.2. A limit of a functor is a universal cone. i.e. A limit of a functor F : D → C is a cone (L, (pD)D∈D) on F such that, for every cone (M, (qD)D∈D) on F, there exists a unique morphism m : M → L such that for every object D ∈ D, qD = pD ◦m. Kan extensions also generalize another important categorical structure: the adjunction.