Adjoint functors are powerful tools in category theory, linking categories through paired functors. They establish a natural bijection between hom-sets, creating a deep connection between objects and morphisms in different categories.

Left adjoints preserve colimits, while right adjoints preserve limits. This property makes adjoint functors crucial for understanding how structures transfer between categories, playing a key role in many mathematical and computational contexts.

Adjoint Functors

Definition of adjoint functors

• Pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ between categories $\mathcal{C}$ and $\mathcal{D}$ that establish a bijection (natural isomorphism) between hom-sets $\text{Hom}{\mathcal{D}}(F(C), D) \cong \text{Hom}{\mathcal{C}}(C, G(D))$ for every pair of objects $C \in \mathcal{C}$ and $D \in \mathcal{D}$
  • Bijection is natural in both $C$ and $D$, meaning it is compatible with morphisms in $\mathcal{C}$ and $\mathcal{D}$
• $F$ is the left adjoint functor and $G$ is the right adjoint functor
• Adjoint functors come with natural transformations $\eta: 1_{\mathcal{C}} \Rightarrow GF$ (unit) and $\varepsilon: FG \Rightarrow 1_{\mathcal{D}}$ (counit) satisfying triangle identities, which express the compatibility between the unit, counit, and the functors
• Left adjoints preserve colimits (coproducts, coequalizers) while right adjoints preserve limits (products, equalizers)

Examples in various categories

• Category of sets ($\mathbf{Set}$)
  • Free functor $F: \mathbf{Set} \to \mathbf{Grp}$ is left adjoint to forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$, constructing the free group on a set and forgetting the group structure
  • Powerset functor $\mathcal{P}: \mathbf{Set} \to \mathbf{Set}^{\text{op}}$ is left adjoint to itself, mapping a set to its powerset and vice versa
• Category of topological spaces ($\mathbf{Top}$)
  • Functor $|-|: \mathbf{Top} \to \mathbf{Set}$ forgetting the topology is right adjoint to discrete topology functor $D: \mathbf{Set} \to \mathbf{Top}$, equipping a set with the discrete topology
  • Functor $|-|: \mathbf{Top} \to \mathbf{Set}$ is left adjoint to indiscrete topology functor $I: \mathbf{Set} \to \mathbf{Top}$, equipping a set with the indiscrete topology
• Category of vector spaces ($\mathbf{Vect}_K$) over field $K$
  • Tensor product functor $- \otimes_K V: \mathbf{Vect}_K \to \mathbf{Vect}_K$ is left adjoint to hom functor $\text{Hom}_K(V, -): \mathbf{Vect}_K \to \mathbf{Vect}_K$ for any vector space $V$, capturing the adjunction between tensor product and linear maps

Left vs right adjoint functors

• If $F: \mathcal{C} \to \mathcal{D}$ is left adjoint to $G: \mathcal{D} \to \mathcal{C}$, then $G$ is right adjoint to $F$, denoted $F \dashv G$
• Unit $\eta: 1_{\mathcal{C}} \Rightarrow GF$ and counit $\varepsilon: FG \Rightarrow 1_{\mathcal{D}}$ satisfy triangle identities $(\varepsilon F) \circ (F \eta) = 1_F$ and $(G \varepsilon) \circ (\eta G) = 1_G$, expressing the compatibility between the unit, counit, and the functors
• Adjoint functors are unique up to natural isomorphism, meaning if $F \dashv G$ and $F \dashv G'$, then $G \cong G'$

Preservation of categorical structures

• Left adjoint functors preserve colimits
  • For diagram $D: \mathcal{J} \to \mathcal{C}$, left adjoint $F: \mathcal{C} \to \mathcal{D}$ satisfies $F(\text{colim } D) \cong \text{colim } (F \circ D)$
  • Preserves coproducts and coequalizers
• Right adjoint functors preserve limits
  • For diagram $D: \mathcal{J} \to \mathcal{D}$, right adjoint $G: \mathcal{D} \to \mathcal{C}$ satisfies $G(\text{lim } D) \cong \text{lim } (G \circ D)$
  • Preserves products and equalizers
• Adjoint functors preserve exponential objects in cartesian closed categories
  • If $F: \mathcal{C} \to \mathcal{D}$ is left adjoint to $G: \mathcal{D} \to \mathcal{C}$ and $\mathcal{C}$, $\mathcal{D}$ are cartesian closed, then $F(C^A) \cong (FC)^{(GA)}$ for objects $A, C \in \mathcal{C}$