nLab
left adjoint

Idea

The left adjoint of a functor, if it exists, is one of two best approximations to a weak inverse of that functor. (The other best approximation is the functor's right adjoint, if it exists.) Note that a weak inverse itself, if it exists, must be a left adjoint.

A left adjoint to a forgetful functor is called a free functor; in general, left adjoints may be thought of as being defined freely, consisting of anything that an inverse might want, regardless of whether it works.

The concept generalises immediately to enriched categories and in 2-categories.

Definitions

Given posets (or prosets) $C$ and $D$ and a monotone function $U:C\to D$, a left adjoint of $U$ is a monotone function $F:D\to C$ such that

$$F(x)\le y\iff x\le U(y)$$F(x) \leq y \;\Leftrightarrow\; x \leq U(y)

for all $x$ in $C$ and $y$ in $D$.

Given locally small categories $C$ and $D$ and a functor $U:C\to D$, a left adjoint of $U$ is a functor $F:D\to C$ with a natural isomorphism between the hom-set functors

$${\mathrm{Hom}}_C(F(-),-),{\mathrm{Hom}}_D(-,U(-)):C^{\mathrm{op}}\times D\to \mathrm{Set}.$$Hom_C(F(-),-), Hom_D(-,U(-)): C^op \times D \to Set .

Given $V$-enriched categories $C$ and $D$ and a $V$-enriched functor $U:C\to D$, a left adjoint of $U$ is a $V$-enriched functor $F:D\to C$ with a $V$-enriched natural isomorphism? between the hom-object functors

$${\mathrm{Hom}}_C(F(-),-),{\mathrm{Hom}}_D(-,U(-)):C^{\mathrm{op}}\times D\to \mathrm{Set}.$$Hom_C(F(-),-), Hom_D(-,U(-)): C^op \times D \to Set .

Given categories $C$ and $D$ and a functor $U:C\to D$, a left adjoint of $U$ is a functor $F:D\to C$ with natural transformations

$$\iota :{\mathrm{id}}_C\to F;U,ϵ:U;F\to {\mathrm{id}}_D$$\iota: id_C \to F ; U,\; \epsilon: U ; F \to id_D

(where $F;U$ etc gives the composite in the forwards, anti-Leibniz order) satisfying certain triangle identities.

Given a 2-category $ℬ$, objects $C$ and $D$ of $ℬ$, and a morphism $U:C\to D$ in $ℬ$, a left adjoint of $U$ is a morphism $F:D\to C$ with $2$-morphisms

$$\iota :{\mathrm{id}}_C\to F;U,ϵ:U;F\to {\mathrm{id}}_D$$\iota: id_C \to F ; U,\; \epsilon: U ; F \to id_D

satisfying the triangle identities.

Although it may not be immediately obvious, these definitions are all compatible.

Whenever $F$ is a left adjoint of $U$, we have that $U$ is a right adjoint of $F$.

Further remarks

• See Galois connection for left adjoints of monotone functions.
• See adjoint functor for left adjoints of functors.
• See adjunction for left adjionts in $2$-categories.
• See examples of adjoint functors for examples.