Left adjoints preserve colimits; right adjoints preserve limits
Functors F and G are adjoint when there is a natural bijection between morphisms FA → B and A → GB. Almost every construction met so far is half of an adjunction — free and forgetful, tensor and Hom, restriction and coinduction. The payoff is immediate: left adjoints preserve colimits, hence are right exact, and right adjoints preserve limits, hence are left exact. The exactness properties that generate Ext and Tor are consequences of adjointness, not coincidences.
Learning objectives
- State the adjunction bijection and its naturality.
- Identify the unit and counit and the triangle identities.
- Write the tensor-hom adjunction and read off exactness.
- Explain why adjoints preserve limits or colimits.
- Use adjointness to transport projectives and injectives.
Section 01The definition
F is the left adjoint, G the right adjoint. Equivalently, there are natural transformations η: 1 → GF (unit) and ε: FG → 1 (counit) satisfying the triangle identities.
| Left adjoint | Right adjoint | Consequence |
|---|---|---|
| Free module on a set | Forgetful to sets | Free modules are projective |
| M ⊗Λ − | Homℤ(M, −) | Tensor is right exact; Hom is left exact |
| Restriction of scalars | HomΛ(Λ′, −) (coinduction) | Produces enough injectives |
| Extension of scalars Λ′ ⊗ − | Restriction of scalars | Change-of-rings spectral sequences |
| Abelianisation | Inclusion of abelian groups | H1(G) is the abelianisation |
| Left Kan extension | Restriction along a functor | Homology of small categories |
Section 02Exactness for free
- Let F be left adjoint to G, and let A → B → C → 0 be exact.
- For any object X, Hom(FC, X) ≅ Hom(C, GX) by adjointness.
- Hom(−, GX) is left exact and contravariant, so it carries the exact sequence to an exact sequence 0 → Hom(C, GX) → Hom(B, GX) → Hom(A, GX).
- Transporting back, 0 → Hom(FC, X) → Hom(FB, X) → Hom(FA, X) is exact for every X. Naturality is what allows the transport.
- Since this holds for all X, FA → FB → FC → 0 is exact.
Left adjoints preserve all colimits — coproducts, cokernels, pushouts, directed colimits. Right adjoints preserve all limits. Right exactness and left exactness are the special cases visible in an abelian category.
Section 03Tensor-hom and its uses
Tensor is right exact
It is a left adjoint, so it preserves cokernels. Its failure of left exactness is measured by Tor.
Adjoints preserve projectives and injectives
If the right adjoint is exact, the left adjoint preserves projectives; if the left adjoint is exact, the right adjoint preserves injectives. This is how enough injectives is proved.
Change of rings
Restriction and extension of scalars are adjoint, and comparing their derived functors produces the change-of-rings spectral sequences.
If a left adjoint exists it is unique up to natural isomorphism, so ‘the’ left adjoint is well defined. This is the same uniqueness argument as for universal properties, and indeed an adjunction is a family of universal properties.
ReferenceFrequently asked questions
Does every functor have an adjoint?
No. The adjoint functor theorem gives conditions — preservation of limits plus a solution set condition — but they are genuine hypotheses. Many natural functors have an adjoint on one side only.
What do the triangle identities say?
That the unit followed by the counit, suitably whiskered, is the identity on each side. They are what make the bijection coherent, and checking them is the standard way to verify a proposed adjunction.
Why does adjointness matter for computing Ext?
Because it lets a computation be moved to a more convenient category. Change-of-rings results and the Grothendieck spectral sequence both rest on an adjunction, and both are used to reduce an intractable Ext to a computable one.
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