Adjoint Functor Theorems

Summary

The General Adjoint Functor Theorem (GAFT) says: a continuous functor (preserves all small limits) from a complete locally small category has a left adjoint if and only if it satisfies the solution-set condition. The Special Adjoint Functor Theorem (SAFT) replaces the solution set with a “well-powered” and “co-well-powered” condition. These are the primary tools for proving that a functor has an adjoint without constructing it explicitly.

Overview

The adjoint functor theorems answer the question: “when does a functor have an adjoint?” The answer involves two ingredients: the functor must respect the categorical structure (preserve limits for a right adjoint), and it must satisfy a set-theoretic smallness condition (the solution-set condition or its variants).

Main Content

The Adjoint Functor Theorem for Preorders

As a warmup, the GAFT has a clean version for preorders:

Proposition 6.3.7: AFT for Preorders (BCT, Ch. 6.3)

Let $P$ be a poset (partial order) in which all meets (infima) exist. A monotone map $G:P\to Q$ has a left adjoint if and only if it preserves all meets.

This is the “classical” Galois connection theorem and motivates the general case.

General Adjoint Functor Theorem (GAFT)

Definition: Solution-Set Condition

A functor $G:\mathcal{B}\to \mathcal{A}$ satisfies the solution-set condition if for each $A\in \mathcal{A}$, there is a small set $\mathcal{S}$ of maps $(A\to G{B}_s{)}_{s\in \mathcal{S}}$ such that every map $A\to GB$ factors as $A\to G{B}_s\to GB$ for some $s\in \mathcal{S}$.

I.e., the comma category $(A\downarrow G)$ has a small weakly initial set of objects.

Theorem 6.3.10: General Adjoint Functor Theorem (BCT, Ch. 6.3)

Let $\mathcal{B}$ be a locally small complete category, and $G:\mathcal{B}\to \mathcal{A}$ a functor. Then $G$ has a left adjoint if and only if:

1. $G$ preserves all small limits (i.e., $G$ is continuous), and
2. $G$ satisfies the solution-set condition.

Proof sketch:

• ($\Rightarrow$): If $F⊣G$, then $G$ preserves limits by ^adjoints-limits. The solution set for $A$ is $\{{\eta }_A:A\to GFA\}$ — a single element.
• ($\Leftarrow$): For each $A$, use the solution set to construct a small diagram, then take the limit of the induced diagram to get the initial object of $(A\downarrow G)$ — which is $FA$ with unit ${\eta }_A$.

The full proof appears in the Appendix of the book.

Example: GAFT for Forgetful Functors (BCT, Ch. 6.3)

The forgetful functor $G:\mathbf{Grp}\to \mathbf{Set}$ is continuous (preserves limits). The solution set for a set $X$ is $\{f:X\to UG\mid G\text{ is a quotient of }{F}_X\}$ where $F_X$ is the free group on $X$ — this is essentially just the free group itself. So $G$ has a left adjoint, which is the free group functor.

Special Adjoint Functor Theorem (SAFT)

Definition: Well-Powered and Cogenerating Set

• A category is well-powered if for each object $A$, the collection of subobjects of $A$ is a small set.
• A cogenerating set for $\mathcal{B}$ is a set $\{C_i{\}}_{i\in I}$ of objects such that for any $f\ne g:A\to B$, there exists $i$ and $h:B\to C_i$ with $h\circ f\ne h\circ g$.

Theorem 6.3.13: Special Adjoint Functor Theorem (BCT, Ch. 6.3)

Let $\mathcal{B}$ be locally small, complete, well-powered, and having a cogenerating set. Then any continuous functor $G:\mathcal{B}\to \mathcal{A}$ has a left adjoint.

The SAFT is “special” because the solution-set condition is automatically satisfied from the structural properties of $\mathcal{B}$.

Applications:

• The categories Set, Ab, Grp, Ring, Top all satisfy the hypotheses of SAFT (or GAFT).
• In particular: any limit-preserving functor from Ab to Ab has a left adjoint.

Freyd’s Original Formulation

Leinster’s SAFT follows Freyd (1964). The theorem says that in a “nice” category, continuity is equivalent to having an adjoint — no extra conditions needed beyond the structural ones already satisfied by the category.

Why “Special”?

The SAFT applies when $\mathcal{B}$ has nice properties (well-powered + cogenerating set), whereas the GAFT is more general but requires verifying the solution-set condition explicitly. In practice:

• Use SAFT when $\mathcal{B}$ is a familiar category like Set or Ab.
• Use GAFT when $\mathcal{B}$ is more exotic or when you need the solution set explicitly.

Connections

• Adjoints and limits (Adjoints and Limits): the necessity direction of GAFT (right adjoints preserve limits).
• Initial objects (Adjunctions via Initial Objects): the construction of the adjoint in GAFT produces initial objects in comma categories.
• The density theorem (^density-theorem) is used in the appendix proof of GAFT.

See Also

• Adjoint Functors — Adjunction definition
• Adjoints and Limits — Right adjoints preserve limits (necessity in GAFT)
• Adjunctions via Initial Objects — The proof constructs initial objects
• Algebras for a Monad - Eilenberg-Moore and Kleisli — limits/colimits of algebras & adjoint construction