Adjoint functor

A concept expressing the universality and naturalness of many important mathematical constructions, such as a free universal algebra, various completions, and direct and inverse limits.

Let $ F : \mathfrak K \rightarrow \mathfrak C $ be a covariant functor in one argument from a category $ \mathfrak K $ into a category $ \mathfrak C $. $ F $ induces a functor

$$ H ^ {F} ( X , Y ) = \ H _ {\mathfrak C } ( F (X) , Y ) : \mathfrak K ^ {*} \times \mathfrak C \rightarrow \mathfrak S , $$

where $ \mathfrak K ^ {*} $ is the category dual to $ \mathfrak K $, $ \mathfrak S $ is the category of sets, and $ H _ {\mathfrak C} ( X , Y ) : \mathfrak K ^ {*} \times \mathfrak C \rightarrow \mathfrak S $ is the basic set-valued functor. The functor $ H ^ {F} $ is contravariant in the first argument and covariant in the second. Similarly, any covariant functor $ G : \mathfrak C \rightarrow \mathfrak K $ induces a functor

$$ H _ {G} ( X , Y ) = \ H _ {\mathfrak K } ( X , G (Y) ) : \ \mathfrak K ^ {*} \times \mathfrak C \rightarrow \mathfrak S , $$

which is also contravariant in the first argument and covariant in the second. The functors $ F $ and $ G $ are adjoint, or form an adjoint pair, if $ H ^ {F} $ and $ H _ {G} $ are isomorphic, that is, if there is a natural transformation $ \theta : H ^ {F} \rightarrow H _ {G} $ that establishes a one-to-one correspondence between the sets of morphisms $ H _ {\mathfrak C} ( F (X) , Y ) $ and $ H _ {\mathfrak K} ( X , G (Y) ) $ for all objects $ X \in \mathop{\rm Ob} \mathfrak K $ and $ Y \in \mathop{\rm Ob} \mathfrak C $. The transformation $ \theta $ is called the adjunction of $ F $ with $ G $, $ F $ is called the left adjoint of $ G $ and $ G $ the right adjoint of $ F $( this is written $ \theta : F \dashv G $, or simply $ F \dashv G $). The transformation $ \theta ^ {-1} : H _ {G} \rightarrow H ^ {F} $ is called the co-adjunction.

Let $ \theta : F \dashv G $. For all $ X \in \mathop{\rm Ob} \mathfrak K $ and $ Y \in \mathop{\rm Ob} \mathfrak C $, let

$$ \epsilon _ {X} = \theta ( 1 _ {F (X) } ) ,\ \ \eta _ {Y} = \theta ^ {-1} ( 1 _ {G (Y) } ) . $$

The families of morphisms $ \{ \epsilon _ {X} \} $ and $ \{ \eta _ {Y} \} $ define natural transformations $ \epsilon : \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $ and $ \eta : F G \rightarrow \mathop{\rm Id} _ {\mathfrak C} $, called the unit and co-unit of the adjunction $ \theta $. They satisfy the following equations:

$$ G ( \eta _ {Y} ) \epsilon _ {G (Y) } = \ 1 _ {G (Y) } ,\ \ \eta _ {F (X) } F ( \epsilon _ {X} ) = \ 1 _ {F (X) } . $$

In general, a pair of natural transformations $ \phi : \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $ and $ \psi : F G \rightarrow \mathop{\rm Id} _ {\mathfrak C} $ leads to an adjoint pair (or adjunction) if the following equations hold:

$$ G ( \psi _ {Y} ) \phi _ {G (Y) } = \ 1 _ {G (Y) } ,\ \ \psi _ {F(X) } F ( \phi _ {X} ) = \ 1 _ {F (X) } $$

for all objects $ X $ and $ Y $. A natural transformation $ \phi : \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $ is the unit of some adjunction if and only if for any morphism $ \alpha : X \rightarrow G (Y) $ in $ \mathfrak K $ there is a unique morphism $ \alpha ^ \prime : F (X) \rightarrow Y $ in $ \mathfrak C $ such that $ \alpha = G ( \alpha ^ \prime ) \epsilon _ {X} $. This property expresses the fact that $ F (X) $ is a free object over $ X $ with respect to the functor $ G $ in the sense of the following definition. An object $ Y \in \mathop{\rm Ob} \mathfrak C $ together with a morphism $ \epsilon : X \rightarrow G (Y) $ is free over an object $ X \in \mathop{\rm Ob} \mathfrak K $ if every morphism $ \alpha : X \rightarrow G ( Y ^ { \prime } ) $ can be written uniquely in the form $ \alpha = G ( \alpha ^ \prime ) \epsilon $ for some morphism $ \alpha ^ \prime : Y \rightarrow Y ^ { \prime } $. A functor $ G : \mathfrak C \rightarrow \mathfrak K $ has a left adjoint if and only if for every $ X \in \mathop{\rm Ob} \mathfrak K $ there is an object $ Y $ that is free over $ X $ with respect to $ G $.

Examples of adjoint functors.

1) If $ G : \mathfrak C \rightarrow {\mathfrak S } $, where $ \mathfrak S $ is the category of sets, then $ G $ has a left adjoint only if it is representable. A representable functor $ G \simeq H ^ {A} = H _ {\mathfrak C} ( A , Y ) $ has a left adjoint if and only if all co-products $ \amalg _ {x \in X } A _ {x} $ exist in $ \mathfrak C $, where $ X \in \mathop{\rm Ob} \mathfrak S $ and $ A _ {x} = A $ for all $ x \in X $.

2) In the category $ \mathfrak S $ of sets, for any set $ A $ the basic functor $ H ^ {A} (Y) = H ( A , Y ) $ is the right adjoint of the functor $ X \times A $.

3) In the category of Abelian groups, the functor $ \mathop{\rm Hom} ( A , Y ) $ is the right adjoint of the functor $ X \otimes A $ of tensor multiplication by $ A $, and the imbedding functor of the full subcategory of torsion groups is the left adjoint of the functor of taking the torsion part of any Abelian group.

4) Let $ P : \mathfrak A \rightarrow \mathfrak S $ be the forgetful functor from an arbitrary variety of universal algebras into the category of sets. The functor $ P $ has a left adjoint $ F : \mathfrak S \rightarrow \mathfrak A $, which assigns to every set $ X $ the free algebra of the variety $ \mathfrak A $ with $ X $ as set of free generators.

5) The imbedding functor $ \mathop{\rm Id} _ {\mathfrak C , \mathfrak K } : \mathfrak C \rightarrow \mathfrak K $ of an arbitrary reflective subcategory $ \mathfrak C $ of a category $ \mathfrak K $ is the right adjoint of the $ \mathfrak C $- reflector (cf. also Reflexive subcategor). In particular, the imbedding functor of the category of Abelian groups in the category of groups has a left adjoint, which assigns to every group $ G $ its quotient group by the commutator subgroup.

Properties of adjoint functors.

The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors. Left adjoints commute with co-limits (e.g. co-products) and send null objects and null morphism into null objects and null morphisms, respectively.

Let $ \mathfrak K $ and $ \mathfrak C $ be categories that are complete on the left and locally small on the left. A functor $ G : \mathfrak G \rightarrow \mathfrak K $ has a left adjoint $ F : \mathfrak K \rightarrow \mathfrak C $ if and only if the following conditions hold: a) $ G $ commutes with limits; b) for every $ X \in \mathop{\rm Ob} \mathfrak K $, at least one of the sets $ H ( X , G (Y) ) $, $ Y \in \mathop{\rm Ob} \mathfrak K $, is non-empty; and c) for every $ X \in \mathop{\rm Ob} \mathfrak K $, there is a set $ S \subset \mathop{\rm Ob} \mathfrak C $ such that every morphism $ \alpha : X \rightarrow G (Y) $ is representable in the form $ \alpha = G ( \alpha ^ \prime ) \phi $, where $ \phi : X \rightarrow G (B) $, $ B \in S $, $ \alpha ^ \prime : B \rightarrow Y $.

By passing to dual categories, one may establish a duality between the concepts of a "left adjoint functor" and a "right adjoint functor" ; this enables one to deduce the properties of right adjoints from those of left adjoints.

The concept of an adjoint functor is directly connected with the concept of a triple (or monad) in a category.

References

[1]	M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)
[2]	S. Maclane, "Categories for the working mathematician" , Springer (1971)

Comments

A category is called complete on the left if small diagrams have limits. A category is called locally small on the left if it has small hom-sets. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.