Isbell Duality

Definition

Let $\mathcal{V}$ be a complete and complete closed symmetric monoidal category and $\mathcal{C}$ a small $\mathcal{V}$-enriched category. There are two enriched functor categories

  1. $[\mathcal{C}^{\operatorname{op}},\mathcal{V}]$, the category of presheaves on $\mathcal{C}$ valued in $\mathcal{V}$.
  2. $[\mathcal{C},\mathcal{V}]$, the category of functors or copresheaves on $\mathcal{C}$ valued in $\mathcal{V}$.
Theorem(Isbell duality) . There is a $\mathcal{V}$-adjunction $$ (\mathcal{O} \dashv \operatorname{Spec}) \colon [\mathcal{C}, \mathcal{V}]^{\operatorname{op}} \overset{\mathcal{O}}{\underset{\operatorname{Spec}}{\leftrightarrows}} [\mathcal{C}^{\operatorname{op}}, \mathcal{V}] $$ between the opposite of the category of copresheaves and the category of presehves, on $\mathcal{C}$ and valued in $\mathcal{V}$. Where, for $X \in [\mathcal{C}^{\operatorname{op}},\mathcal{V}]$ $$ \mathcal{O}(X) \colon c \mapsto [\mathcal{C}^{\operatorname{op}},\mathcal{V}](X,\mathcal{C}(\cdot,c)) $$ and for $A\in [\mathcal{C},\mathcal{V}]^{\operatorname{op}}$ $$ \operatorname{Spec}(A) \colon c \mapsto [\mathcal{C}, \mathcal{V}]^{\operatorname{op}}(\mathcal{C}(c,\cdot),A). $$

Duality between geometry and algebra

The category of presheaves $[\mathcal{C}^{\operatorname{op}},\mathcal{V}]$ is often the category for higher geometry, while the copresheaves $[\mathcal{C},\mathcal{V}]^{\operatorname{op}}$ is that for higher algebra.

Restrict to the case when $\mathcal{V}$ is $\mathsf{Set}$,

Note that

More abstractly, the category $\mathcal{C}$ should be seen as a theory-encoding entity, or more precisely the syntactic category of a theory. Its realization in a model forms a category $[\mathcal{C},\mathcal{V}]$, of which the opposite is the copresheaves. The category $\mathcal{C}$ is a prototype/syntax of the space $[\mathcal{C}^{\operatorname{op}},\mathcal{V}]$. The actual spaces $X\in [\mathcal{C}^{\operatorname{op}},\mathcal{V}]$ are realizations of the space in the enriching (or ambient) category $\mathcal{V}$. Examples: