Isbell conjugacy

Isbell conjugacy (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.^[1]

Definition

Let $\mathcal{V}$ be a symmetric monoidal closed category, and let $\mathcal{A}$ be a small category enriched in $\mathcal{V}$.

The Isbell conjugacy is an adjunction between the categories $\mathcal{V}^{\mathcal{A}^{op}}$ and $(\mathcal{V}^{\mathcal{A}})^{op}$ arising from the Yoneda embedding $Y:\mathcal{A}\rightarrow\mathcal{V}^{\mathcal{A}^{op}}$ and the dual Yoneda embedding $Z:\mathcal{A}\rightarrow(\mathcal{V}^{\mathcal{A}})^{op}$.

References

1. Lawvere, William (1986), Taking Categories Seriously

Bibliography

• Kelly, Gregory Maxwell (1982), Basic Concepts in Enriched Category Theory
• Day, Brian; Lack, Stephen (2006), Limits of Small Functors