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

Lawvere, F. W. (1986), "Taking categories seriously", Revista Colombiana de Matemáticas, 20 (3–4): 147–178, MR 0948965

Bibliography

Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, MR 0651714.
Day, Brian J.; Lack, Stephen (2007), "Limits of small functors", Journal of Pure and Applied Algebra, 210 (3): 651–663, arXiv:math/0610439, doi:10.1016/j.jpaa.2006.10.019, MR 2324597.

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[1] Lawvere, F. W. (1986), "Taking categories seriously", Revista Colombiana de Matemáticas, 20 (3–4): 147–178, MR 0948965