Kan extensions

Builds on

Required prerequisites, all the way down: 24

You will learn to

• Define left and right Kan extensions by universal properties and compute them by pointwise formulas when the target has the necessary colimits or limits.
• Express limits, colimits and adjoint functors as Kan extensions.
• Use left Kan extension along the Yoneda embedding to extend a functor from representables to all presheaves.

Definition

• A left Kan extension of F : C → E along K : C → D is a functor Lan_K F : D → E with a natural transformation F ⇒ (Lan_K F) ∘ K that is universal among such pairs.
• Right Kan extensions are defined dually.

Theorem

• Pointwise formula: (Lan_K F)(d) is the colimit of F over the comma category (K ↓ d), when these colimits exist.
• The limit of F : J → C is the right Kan extension of F along the unique functor from J to the terminal category.
• A functor G has a left adjoint if and only if the right Kan extension of the identity along G exists and is preserved by G; that Kan extension is then the left adjoint.

Common misconceptions

• Thinking Kan extensions always exist; pointwise ones do when the source category is small, the category extended along is locally small, and the target has enough colimits (or limits).
• Confusing left and right Kan extensions; pointwise left Kan extensions are computed by colimits.

Optional discoveries

• ∞-categories Kan extensions, limits and adjunctions all have ∞-categorical versions that are central to modern homotopy theory.
• ← Universal properties

Further reading

• Saunders Mac Lane, Categories for the Working Mathematician (1971)
• Emily Riehl, Category Theory in Context (2016)