I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\dashv\Gamma$ for schemes through Isbell duality. As I understand it, Isbell duality relates appropriate "algebraic" and "geometric" categories via the following construction (see 1).
Let $\mathcal{A}$ be a (coextensive by 1, so $\mathcal{A}^\text{op}$ is extensive) category, and $\mathcal{B}$ a (Cartesian, to form ring objects) category. If there exists a dualizing object $\mathbb{A}$ for $\mathcal{A}$ and $\mathcal{B}$, consider the functors $$\underline{\operatorname{hom}}_{\mathcal{B}}(-,\mathbb{A}):\mathcal{B}\longrightarrow\mathcal{A}^\text{op}$$
and
$$\underline{\operatorname{hom}}_{\mathcal{A}}(-,\mathbb{A}):\mathcal{A}^\text{op}\longrightarrow\mathcal{B}.$$
For "nice enough" categories, we can realize an adjunction $\underline{\operatorname{hom}}_{\mathcal{B}}(-,\mathbb{A})\dashv\underline{\operatorname{hom}}_{\mathcal{A}}(-,\mathbb{A})$ by taking the unit $\eta$ and counit $\epsilon$ as evaluation. This construction induces an equivalence of categories between those $X$ for which $\eta_X$ is an isomorphism and $A$ for which $\epsilon_A$ is an isomorphism.
This construction (pulling from 2) allows us to recover Gelfand duality $C^*\mathsf{Alg}_{\text{com}}^\text{op}\leftrightarrows\mathsf{Top}_\text{cpt}$, the equivalence of affine varieties over $k=\overline{k}$ and finitely generated integral domains (over $k$), and Stone duality, among others.
My question stems from the following realization. In each of these cases, the dualizing object in $\mathcal{B}$ is realized as the internalization of an "$\mathcal{A}$-type object" to $\mathcal{B}$. Indeed, Gelfand duality stems from considering the $\mathbb{C}$-algebra object $\mathbb{A}=\mathbb{C}$ in $\mathsf{Top}$, Stone duality comes from regarding the Sierpinski space $\{0,1\}$ as a frame, and our "affine-geometric duality" follows from regarding $\mathbb{A}^1_k$ as a ring object in $\mathsf{Aff}_k$.
However, in the category of schemes, there does not seem to be a "natural" commutative ring to consider dual (that is, to represent the functor $\mathsf{CRing}^\text{op}\longrightarrow \mathsf{Sch}$) to the affine line $\mathbb{A}^1=\operatorname{Spec}(\mathbb{Z}[x])$, which of course we can equip with a ring object structure (see 3 for more details).
Thus, is there a way that we can see the adjunction $\operatorname{Spec}\dashv\Gamma$ through Isbell duality, without appealing to dualizing objects? If possible, I'd love to see how this duality is expressed for (complex) analytic spaces. Thank you for any help or clarification.
References:
1: Why is there a duality between spaces and commutative algebras?
3: Dualizing object in the duality between commutative rings and affine schemes
Best Answer
Stone Spaces by Peter Johnstone (CUP 1982) is about just this subject.
It is an exceptionally well written book. It would be better that you read it than that I make any attempt to summarise it for you.
By "exceptionally" I mean in contrast to the usual way in which research level pure maths books are written, namely by making it clear before the end of the first page that you are not welcome as a reader unless you are a grad student of the author.
Stone Spaces was written to sell locale theory (thought Peter would never use such a vulgar word) to the kind of mathematicians who define "filters" or "ideals" in their structures and then put a topology on the set of them. He demonstrates with several examples how you get the locale directly from the algebra and that Choice is only needed to define "points" of it, but that these are not really needed at all.