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