The universal way to extend a functor
Given K: A → B and T: A → C, a Kan extension is the best approximation to a functor B → C restricting to T. Left and right versions are the left and right adjoints of restriction along K, and both are computed by colimits and limits over comma categories. Adjoints, limits and induced representations are all instances, and in this stream the left Kan extension is what defines the homology of a small category.
Learning objectives
- State the universal property of the left and right Kan extensions.
- Write the colimit and limit formulas.
- Identify the adjunction with restriction.
- Recognise familiar constructions as Kan extensions.
Section 01The universal property
Restriction along K gives K*: Fun(B, C) → Fun(A, C). Its left and right adjoints, when they exist, are the left and right Kan extensions:
So a natural transformation out of LanKT is exactly a natural transformation out of T after restriction, and dually.
Mac Lane's slogan. Limits and colimits are Kan extensions along the functor to the one-object category; adjoints are Kan extensions along an identity; induced representations are Kan extensions along a subgroup inclusion. Recognising the pattern collapses several constructions into one.
Section 02Standard instances
| Construction | Kan extension along |
|---|---|
| Colimit of a diagram | The functor from the index category to the terminal category |
| Limit of a diagram | The same, right version |
| Induced module Λ ⊗Γ − | The inclusion of a subring, left version |
| Coinduced module HomΓ(Λ, −) | The same, right version |
| Left adjoint of a functor | That functor, applied to the identity |
| Homology of a small category | The functor to the terminal category, left derived |
| Sheafification | A Kan extension followed by a localisation |
The colimit formula is taken over the objects of A equipped with a map from their image to b — the comma category K ↓ b. That indexing is what makes the construction universal rather than ad hoc, and it is why existence requires the target to have enough colimits.
Section 03Existence and exactness
- Stage 01Pointwise existenceIf C has all small colimits and A is small, LanKT exists and is computed pointwise by the formula.
- Stage 02Left adjoint behaviourBeing a left adjoint, LanK preserves colimits and is right exact in an abelian setting.
- Stage 03Derived versionWhen the target is abelian, LanK can be derived; its left derived functors are the homology of the relevant comma categories.
- Stage 04Grothendieck sequenceComposing Kan extensions gives a Grothendieck spectral sequence relating the derived functors of the pieces.
K*LanKT is not generally isomorphic to T — only when K is fully faithful. The word ‘extension’ is therefore slightly optimistic in the general case, and assuming the restriction returns T is a real error.
ReferenceFrequently asked questions
Do Kan extensions always exist?
Not always. Pointwise existence needs the target to have the relevant colimits or limits and the source to be small. Without those hypotheses a Kan extension may fail to exist, or exist without being computed by the formula.
What is the connection with induced representations?
For a subgroup inclusion viewed as a functor of one-object categories, the left Kan extension is induction and the right is coinduction. Shapiro's lemma is then the statement that these adjoints are exact enough to commute with taking derived functors.
Why are Kan extensions relevant to homological algebra?
Because deriving them produces homology theories. The homology of a small category, the homology of a group and the homology of a simplicial object are all left derived Kan extensions along a suitable functor.
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