Isbell Duality

TL;DR

The paper explores Isbell duality, a fundamental concept in category theory, illustrating an adjunction between presheaves and copresheaves on any category, highlighting its mathematical elegance.

Contribution

It provides a clear explanation of Isbell duality and demonstrates its application as an adjunction between presheaves and copresheaves for any category.

Findings

Illustrates Isbell duality with examples

Establishes the adjunction between presheaves and copresheaves

Highlights the mathematical beauty of the duality

Abstract

Mathematicians love dualities. After a brief explanation of dualities, with examples, we turn to one of the purest and most beautiful: Isbell duality. For any category $\mathsf{C}$, this gives an adjunction between the category of presheaves on $\mathsf{C}$, namely the functor category $[\mathsf{C}^{\text{op}}, \mathsf{Set}]$, and the opposite of the category of copresheaves on $\mathsf{C}$, namely $[\mathsf{C}, \mathsf{Set}]^{\text{op}}$.