For presheaf $\mathcal{F}$ on space $X$, sheafification is left Kan extension of $\mathcal{F}$ (viewed on basis) along inclusion of basis into all open sets. This extends presheaf to sheaf universally.

For $R$-modules:

• $M \otimes_R -$ is left Kan extension of evaluation functor along $\text{Hom}_R(M, R)$
• $\text{Hom}_R(M, -)$ is right Kan extension

This explains tensor-hom adjunction via Kan extensions.

For group homomorphism $\phi: H \to G$:

• Induced representation $\text{Ind}_H^G$ is left Kan extension
• Restricted representation $\text{Res}^G_H$ is restriction functor

Frobenius reciprocity follows from Kan extension universal property.

For continuous $f: X \to Y$:

• Direct image $f_*: \textbf{Sh}(X) \to \textbf{Sh}(Y)$ is right Kan extension
• Inverse image $f^*: \textbf{Sh}(Y) \to \textbf{Sh}(X)$ is left adjoint

The adjunction $f^* \dashv f_*$ is instance of Kan extension adjunction.

For presheaf $P: \mathcal{C}^{\text{op}} \to \textbf{Set}$, left Kan extension along Yoneda embedding $\mathcal{Y}: \mathcal{C} \to [\mathcal{C}^{\text{op}}, \textbf{Set}]$ computes colimit: $\text{Lan}_\mathcal{Y} P = \text{colim}_{(A, x) \in \int P} \mathcal{Y}(A)$

Geometric realization $|-|: \textbf{sSet} \to \textbf{Top}$ is left Kan extension of standard simplex functor $\Delta \to \textbf{Top}$ along Yoneda. This explains why it's left adjoint to singular complex.

For monoidal category $\mathcal{C}$, Day convolution on $[\mathcal{C}^{\text{op}}, \textbf{Set}]$ defined using Kan extension of tensor product. This makes presheaf category monoidal.