Kan Extensions

Quoting nCatLab on Kan Exensions:

Idea

The Kan extension of a functor F : C → D with respect to a functor

    C
    |
    | p
    ↓
    C'

is, if it exists, a kind of best approximation to the problem of finding a functor C' → D such that

        F
    C -----> D
    |       ↗
    | p   /
    |   /
    ↓ /
    C'

hence to extending the domain of F through p from C to C'. More generally, this makes sense not only in Cat but in any 2-category. Similarly, a Kan lift is the best approximation to lifting a morphism F : C → D through a morphism

    D'
    |
    ↓
    D

to a morphism F̂

                D'
              ↗ |
            /   |
        F̂ /     |
        /       |
      /   F     ↓
    C --------> D

Kan extensions are ubiquitous. See the discussion at Examples below.

Definitions

There are various slight variants of the definition of Kan extension. In good cases they all exist and all coincide, but in some cases only some of these will actually exist. We (have to) distinguish the following cases:

• “ordinary” or “weak” Kan extensions
  These define the extension of an entire functor, by an adjointness relation.
  Here we (have to) distinguish further between
  • global Kan extensions,
    which define extensions of all possible functors of given domain and codomain (if all of them indeed exist);
  • local Kan extensions,
    which define extensions of single functors only, which may exist even if not every functor has an extension.
• “pointwise” or “strong” Kan extensions
  These define the value of an extended functor on each object (each “point”) by a weighted (co)limit.
  Furthermore, a pointwise Kan extension can be “absolute”.

If the pointwise version exists, then it coincides with the “ordinary” or “weak” version, but the former may exist without the pointwise version existing. See below for more. Some authors (such as Kelly) assert that only pointwise Kan extensions deserve the name “Kan extension,” and use the term as “weak Kan extension” for a functor equipped with a universal natural transformation. It is certainly true that most Kan extensions which arise in practice are pointwise. This distinction is even more important in enriched category theory.

Ordinary or weak Kan extensions

Global Kan extensions

Let p : C → C' be a functor. For D any other category, write p× : (C' → D) → (C → D) for the induced functor on the functor categories: this sends a functor h : C' → D to the composite functor

                p      h
      p* h : C --> C' --> D

Pullback-along functor

Definition. If p× has a left adjoint, typically denoted p_! : (C → D) → (C' → D) or Lan_p : (C → D) → (C' → D) then this left adjoint is called the (ordinary or weak) left Kan extension operation along p. For h ∈ (C → D) we call p_! h the left Kan extension of h along p. Similarly, if p× has a right adjoint, this right adjoint is called the right Kan extension operation along p. It is typically denoted p_× : (C → D) → (C' → D) or Ran = Ran_p : (C → D) → (C' → D). The analogous definition clearly makes sense as stated in other contexts, such as in enriched category theory. Observation. If C' = 1 is the terminal category, then

• the left Kan extension operation forms the colimit of a functor;
• the right Kan extension operation forms the limit of a functor.

Left Kan extensions

Colimits are initial morphisms.

Right Kan extensions

Limits are terminal morphisms