##### Actions

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 $G$.

## Contents

### 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)