Freyd Adjoint Functor Theorem

The Freyd Adjoint Functor Theorem gives a criterion for a functor to have an adjoint. It converts the problem of constructing an adjoint (which involves building universal objects) into checking two conditions: preservation of limits and a set-theoretic "solution set" condition.

Statement

Theorem2.8General Adjoint Functor Theorem (Freyd)

Let $G : \mathcal{D} \to \mathcal{C}$ be a functor where $\mathcal{D}$ is complete and locally small. Then $G$ has a left adjoint if and only if:

$G$ preserves all (small) limits.
Solution set condition: For every object $C \in \mathcal{C}$ , there exists a set $S_C$ of morphisms $\{f_i : C \to G(D_i)\}_{i \in I}$ such that every morphism $f : C \to G(D)$ factors as $f = G(h) \circ f_i$ for some $i \in I$ and some morphism $h : D_i \to D$ in $\mathcal{D}$ .

Theorem2.9Special Adjoint Functor Theorem

Let $G : \mathcal{D} \to \mathcal{C}$ be a functor where $\mathcal{D}$ is complete, locally small, and well-powered, and $\mathcal{D}$ has a cogenerating set. Then $G$ has a left adjoint if and only if $G$ preserves all small limits.

RemarkWhen the solution set condition is automatic

The Special Adjoint Functor Theorem says that for "nice enough" categories (complete, locally small, well-powered, with a cogenerating set), limit preservation alone suffices. The categories $\mathbf{Set}$ , $\mathbf{Ab}$ , $R\text{-}\mathbf{Mod}$ , $\mathbf{Grp}$ , and $\mathbf{Top}$ all satisfy these conditions.

Proof Sketch

ProofProof sketch of the General AFT

Necessity. If $G$ has a left adjoint $F$ , then $G$ preserves limits by RAPL. The solution set condition holds with $S_C = \{\eta_C : C \to GF(C)\}$ (the unit), since every $f : C \to G(D)$ factors as $f = G(\tilde{f}) \circ \eta_C$ by the adjunction.

Sufficiency. We construct the left adjoint $F(C)$ as follows. Consider the category $\mathcal{I}_C$ whose objects are pairs $(D, f : C \to G(D))$ and whose morphisms $(D, f) \to (D', f')$ are maps $h : D \to D'$ with $G(h) \circ f = f'$ . The solution set condition ensures $\mathcal{I}_C$ has a small cofinal subcategory. Since $\mathcal{D}$ is complete, we can take the limit $L = \varprojlim_{(D,f) \in \mathcal{I}_C} D$ . Since $G$ preserves limits, $G(L) = \varprojlim G(D)$ , and the compatible family of $f$ 's gives a map $C \to G(L)$ . Checking the universal property shows $L$ serves as $F(C)$ .

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Examples and Applications

ExampleExistence of free groups

The forgetful functor $U : \mathbf{Grp} \to \mathbf{Set}$ preserves all limits (products of groups are products of underlying sets, etc.) and $\mathbf{Grp}$ satisfies the hypotheses of the Special AFT. Therefore $U$ has a left adjoint: the free group functor.

ExampleExistence of free modules

$U : R\text{-}\mathbf{Mod} \to \mathbf{Set}$ preserves limits and $R\text{-}\mathbf{Mod}$ is complete, locally small, well-powered, with a cogenerating set. The Special AFT gives the existence of free $R$ -modules.

ExampleExistence of Stone-Cech compactification

The inclusion $\iota : \mathbf{CHaus} \hookrightarrow \mathbf{Top}$ preserves limits. Since $\mathbf{CHaus}$ is complete and satisfies the solution set condition (using $[0,1]$ as a cogenerator), the General AFT gives a left adjoint $\beta : \mathbf{Top} \to \mathbf{CHaus}$ : the Stone-Cech compactification.

ExampleExistence of sheafification

The inclusion $\iota : \mathbf{Sh}(X) \hookrightarrow \mathbf{PSh}(X)$ preserves all limits. The AFT gives the left adjoint: the sheafification functor.

ExampleExistence of tensor products

The functor $\mathrm{Hom}_R(M, -) : R\text{-}\mathbf{Mod} \to \mathbf{Ab}$ preserves limits. The AFT provides the left adjoint $M \otimes_R - : \mathbf{Ab} \to R\text{-}\mathbf{Mod}$ (more precisely, via the appropriate reformulation).

ExampleRepresentable functors preserve limits

The representable functor $\mathrm{Hom}(A, -) : \mathcal{C} \to \mathbf{Set}$ preserves all limits. This is an immediate consequence of the definition of limits via universal properties. Conversely, by the AFT, a functor $\mathcal{C} \to \mathbf{Set}$ that preserves all limits and satisfies the solution set condition is representable.

ExampleColimit-preserving functors between presheaf categories

For small categories $\mathcal{C}$ and $\mathcal{D}$ , a cocontinuous (colimit-preserving) functor $F : [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}] \to [\mathcal{D}^{\mathrm{op}}, \mathbf{Set}]$ is determined by its restriction to representables. This is a consequence of the density theorem and the AFT.

ExampleNon-example: no left adjoint to the inclusion of fields

The inclusion $\mathbf{Field} \hookrightarrow \mathbf{CRing}$ does not have a left adjoint. There is no "free field" on a commutative ring: the category of fields is not complete, and the solution set condition fails for fundamental reasons (there is no way to "universally" adjoin inverses to all nonzero elements).

ExampleExistence of limits implies representability (Brown)

In homotopy theory, the Brown representability theorem can be viewed as an analogue of the AFT: a contravariant functor from the homotopy category of CW-complexes to $\mathbf{Set}$ that converts coproducts to products and satisfies a Mayer-Vietoris condition is representable.

ExampleLeft adjoint to restriction of scalars

For a ring homomorphism $\varphi : R \to S$ , restriction of scalars $\varphi^* : S\text{-}\mathbf{Mod} \to R\text{-}\mathbf{Mod}$ preserves all limits. By the AFT, it has a left adjoint: extension of scalars $S \otimes_R - : R\text{-}\mathbf{Mod} \to S\text{-}\mathbf{Mod}$ .

ExampleRight adjoint to restriction of scalars

The same restriction of scalars $\varphi^*$ also preserves colimits (it has an exact left adjoint). By the dual AFT, it also has a right adjoint: coextension of scalars $\mathrm{Hom}_R(S, -) : R\text{-}\mathbf{Mod} \to S\text{-}\mathbf{Mod}$ .

ExampleAdjoint Functor Theorem for abelian categories

In the context of Grothendieck abelian categories, if $G : \mathcal{A} \to \mathcal{B}$ is an exact functor between Grothendieck categories that preserves small products, then $G$ has a left adjoint. The solution set condition is automatically satisfied by the existence of a generator.

Significance

RemarkExistence vs construction

The AFT is an existence theorem: it tells you an adjoint exists without explicitly constructing it. In practice, knowing the adjoint exists is often sufficient, and explicit constructions (free groups, tensor products, etc.) can be developed independently. The theorem is most powerful when applied to prove that certain functors must have adjoints.

RemarkConnection to derived categories

In the Gelfand-Manin framework, the AFT and its variants are used to establish the existence of derived functors and to prove that certain functors between derived categories have adjoints (e.g., the Grothendieck duality adjunction $Rf_* \dashv f^!$ ).