Isbell Duality | Itinerarium Mentis

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

$[\mathcal{C}^{\operatorname{op}},\mathcal{V}]$, the category of presheaves on $\mathcal{C}$ valued in $\mathcal{V}$.
$[\mathcal{C},\mathcal{V}]$, 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 $\mathsf{Set}$,

Geometry: Let $X\in[\mathcal{C}^{\operatorname{op}},\mathsf{Set}]$, given $U\in \mathcal{C}$, the set $X(U)$ is the allowed maps from $U$ into the ‘space’ underlying $X$ (or rather all possible spaces modeled by $X$). This is more explicit when $\mathcal{C}$ is a site. The consistency conditions with the site structure forces the category of spaces to be restricted to the category of sheaves that embbeds full and faithfully into $[\mathcal{C}^{\operatorname{op}},\mathsf{Set}]$. Given a covering $\{U_i \hookrightarrow U\}$ of $U\in\mathcal{C}$, the allowed maps $X(U_i)$ assembles into the allowed maps $X(U)$ and give rise to the notion of ‘subspaces’.
Algebra: Let $A\in [\mathcal{C},\mathsf{Set}]^{\operatorname{op}}$, for each $U\in\mathcal{C}$, the set $A(U)$ is the allowed maps from the space $\operatorname{Spec}(A)$ underlying $A$ into $U$ (notice that $\text{Spec}(A)(U) = [\mathcal{C},\mathcal{V}]^{\operatorname{op}}(\mathcal{C}(U,\cdot),A) = [\mathcal{C},\mathcal{V}](A,\mathcal{C}(U,\cdot))$), hence $A(U)$ is the collection of $U$-valued functions on the ‘space’ $\operatorname{Spec}(A)$.

Note that

$[\mathcal{C}^{\operatorname{op}},\mathsf{Set}]$ is the free cocompletion of $\mathcal{C}$, but also has all limits.
$[\mathcal{C},\mathsf{Set}]^{\operatorname{op}}$ is the free completion of $\mathcal{C}$, but also has all colimits.

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:

$\mathcal{C} = \mathsf{CartSp}$, the category of Euclidean spaces with smooth maps between them. Then $[\mathcal{C}^{\operatorname{op}},\mathsf{Set}]$ subsumes the category of smooth manifolds. See also Diffeological space.
$\mathcal{C} = BG$. the delooping of a group. Then $\mathcal{C}^{\operatorname{op}}=\mathcal{C},\mathsf{Set}$ is the category of $G$-sets.