# Isbell Duality and Dualizing Scheme Objects

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:

• Spec is multi-adjoint to $\hom(-,\mathbb A^1)$. There is a general notion of Diers spectrum that works in many other interesting cases. Jul 24 at 6:38
• @მამუკაჯიბლაძე: What categories are you viewing those functors as going from/to, to view them as a multi-adjunction? Jul 24 at 7:20
• @PeterLeFanuLumsdaine I meant the forgetful functor from local commutative rings and local homomorphisms to commutative rings and homomorphisms. It has a multiadjoint which assigns to a ring $A$ the collection of localizations $(A\to A_{\mathfrak p})_{{\mathfrak p}\in\operatorname{Spec}(A)}$. It is multiadjoint since any homomorphism $A\to L$ to a local ring $L$ factors uniquely through some local homomorphism $A_{\mathfrak p}\to L$ for a unique $\mathfrak p$ Jul 25 at 6:37
• But to be honest, to include all schemes the only approach that I am aware of goes all the way to ringed toposes vs (local ring)ed toposes and invokes the theory of Cole spectra Jul 25 at 6:42

• Thank you for the reference (I'm excited to get my hands on a copy). In the meantime, if you don't mind, could you briefly explain how locales show up in a "geometric" context? Taking a look at nLab, we do get a full subcategory-inclusion by $\operatorname{Sh}(-):\mathsf{Loc}\hookrightarrow\mathsf{Topos}$, so is there any way of connecting this inclusion to ring objects associated to (localic, or possibly arbitrary) topoi? I'm not too familiar with topos theory, so I really appreciate any help. Jul 24 at 17:36