I'm looking for a roadmap to learning about Isbell duality. I know a reasonable amount about several of the "specific" dualities (Gelfand duality, AffSch – CRing, frames – locales, etc), especially affine schemes. However, I'm having trouble finding anything that amounts to a coherent presentation of the general case (with clear prerequisites), and how it ties back to the specific cases.
The nlab article on space and quantity I can follow the "idea" part of, but when they get to the actual Isbell duality, I get lost at symmetric monoidal categories, $V$-enriched categories, ends, coends… and what's more problematic is, I can't find a source that cleanly ties the general case back to the specific cases, or that discuss interesting examples where you have just an adjunction (instead of a full equivalence).
Any guidance or resources are helpful, especially those with prerequisites clearly spelled out. I currently know basic category theory (first few chapters of Categories Work, chapter 1 of Vakil Algebraic Geometry).
Best Answer
Since you are unfamiliar with enriched categories, let me describe the special case of ordinary categories first.
Let $\mathcal{C}$ be an essentially small category, e.g. the opposite of the category of finitely presentable commutative rings. The category $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ has a universal property: it is the free completion of $\mathcal{C}$ by colimits of small diagrams. Dually, $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ is the free completion of $\mathcal{C}$ by limits. It turns out that $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ has limits and $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ has colimits, so the two universal properties applied to the two Yoneda embeddings $\mathcal{C} \to [\mathcal{C}^\textrm{op}, \textbf{Set}]$ and $\mathcal{C} \to [\mathcal{C}, \textbf{Set}]^\textrm{op}$ yield a limit-preserving functor $\operatorname{Spec} : [\mathcal{C}, \textbf{Set}]^\textrm{op} \to [\mathcal{C}^\textrm{op}, \textbf{Set}]$ and a colimit-preserving functor $O : [\mathcal{C}^\textrm{op}, \textbf{Set}] \to [\mathcal{C}, \textbf{Set}]^\textrm{op}$. Isbell duality is the observation that this constitutes an adjunction: $$[\mathcal{C}, \textbf{Set}]^\textrm{op} (O (X), B) \cong [\mathcal{C}^\textrm{op}, \textbf{Set}](X, \operatorname{Spec} B)$$ (This is basically an application of the Yoneda lemma.) Being "free", colimits in $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ behave very nicely and resemble what we expect of colimits in a category of spaces. We can think of $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ as a category of spaces locally modelled by objects in $\mathcal{C}$. On the other hand, we may think of $[\mathcal{C}, \textbf{Set}]$ as a category of algebras modelled by objects in $\mathcal{C}^\textrm{op}$, but the justification for this is not as good, in my opinion.
As a general fact, any adjunction restricts to an equivalence of categories between certain subcategories. By construction, the representable functors on both sides are in those subcategories, but there may be more objects. In general, this adjunction is "just" an adjunction.
One can produce many variations on this theme: you could replace $\textbf{Set}$ with some other category $\mathcal{V}$ and look at $\mathcal{V}$-enriched categories instead of ordinary categories; you could restrict to specified subcategories of $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ or $[\mathcal{C}, \textbf{Set}]^\textrm{op}$, etc. Unfortunately, none of them really capture the classical dualities on the nose – the size restrictions are an unavoidable nuisance, and unless you rig the situation by redefining the objects under study eventually you do have to do some non-trivial work to show that two prima facie different things are the same. The case of affine schemes and commutative rings is especially an example of this – if you use the functor of points definition of scheme the duality is almost a triviality, but if you use the locally ringed space definition of scheme you get a remarkable representation theorem.
If, nonetheless, you want to pursue the subject, I suppose you could first finish reading the rest of CWM, and then read [Kelly, Basic concepts of enriched category theory].