1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety its algebra of regular functions to $\mathbb{A}^1$, and to each finitely generated reduced algebra its (ringed) space of algebra homomorphisms to $\mathbb{C}$. This is almost a tautology, but in some sense a deep one.
2) The category of compact Hausdorff spaces is equivalent to the opposite category of $\mathrm{C}^*$-algebras. The equivalence associates to a compact Hausdorff space its algebra of continuous functions to $\mathbb{C}$ (equipped with the sup norm), and to each $\mathrm{C}^*$-algebra its space of $\mathrm{C}^*$-algebra homomorphisms to $\mathbb{C}$. This is due to Gelfand.
3) The category of totally-disconnected compact Hausdorff spaces is equivalent to the opposite category of Boolean algebras. The equivalence associates to a totally-disconnected compact Hausdorf space its algebra of continuous functions to $\{0,1\}$, and to each Boolean algebra its space of Boolean algebra homomorphisms to $\{0,1\}$. This is due to Stone.
4) The $\infty$-category of simply-connected rational spaces of finite type is equivalent to the opposite $\infty$-category of simply-coconnected coconncetive $\mathbb{E}_\infty$-algebras of finite type over $\mathbb{Q}$. The equivalence associates to a rational space $X$ its cochain complex $C^*(X,\mathbb{Q})$, and to each $\mathbb{E}_\infty$-algebra over $\mathbb{Q}$ its space of $\mathbb{Q}$-linear $\mathbb{E}_\infty$-algebra maps to $\mathbb{Q}$. This is an $\infty$-categorical reformulation due to Lurie of a classical theorem of Sullivan.
5) The $\infty$-category of pro-$p$-finite spaces is equivalent to the opposite category of solvable $\mathbb{E}_\infty$-algebras over $\overline{\mathbb{F}}_p$. The equivalence associates to a pro-$p$-finite space $\{X_i\}_{i \in I}$ the corresponding colimit of cochain complexes $colim_i C^*(X_i,\overline{\mathbb{F}}_p)$. If $X$ is a $p$-finite space then $X$ can be reconstructed as the space of $\overline{\mathbb{F}}_p$-linear $\mathbb{E}_\infty$-algebra maps from $C^*(X,\overline{\mathbb{F}}_p)$ to $\overline{\mathbb{F}}_p$. Here $\overline{\mathbb{F}}_p$ is the algebraic closure of $\mathbb{F}_p$. The theorem in this form is due to Lurie, but a the original version of the theorem (working with $p$-complete spaces) is due to Mandell.
Indeed, if $R$ is a commutative-ring type object in some category of geometric objects, and $X$ is a geometric object carrying a sufficient supply of maps to $R$, then it is worthwhile to look at the ring of functions from $X$ to $R$, as this object will carry a lot of information on $X$. In the other direction, it is often useful to study commutative-ring objects by trying to realize them as rings of functions on some geometric object, their "spectrum". However, even with this in mind, it is still quite striking that this approach yields so many anti-equivalences between full subcategories of geometric objects and full subcategories of commutative-algebraic objects. It is like the geometer/topologist/homotopy-theorist is walking on one continent, and the commutative-algebraist is walking on another, and yet they keep stumbling upon the exact same species of categories (only in reverse).
What is going on here? Is there any conceptual explanation to this phenomenon? Is there any general argument which suggests, even heuristically, that the above equivalences are to be expected?
Best Answer
I don't claim to have a complete answer but here are some miscellaneous comments.
Note that topological spaces are already very nearly defined to be dual to certain commutative algebra-like structures, namely their frames of open subsets. The simplest interesting case of this duality is a duality between finite sets and finite Boolean rings / algebras. One way to think about this conceptually is that the open subsets of a topological space axiomatize verifiable or semidecidable properties of a point in that space: that is, properties such that if they hold you can check that that's true, but such that if they don't hold you can't necessarily check that that's true. See this math.SE question for more discussion on this point. For example, the open sets in the product topology on $\{ 0, 1 \}^{\mathbb{N}}$ correspond precisely to those properties of an infinite sequence of zeroes and ones that you can verify by looking at finitely many terms of the sequence. (Once we agree that verifiable properties are a cool thing to look at, we should also agree that we care about logical operations on them, like AND and OR, and this is where the commutative algebra-like structure comes from.)
One way to define the category of commutative $k$-algebras is as follows. Let $\text{Poly}(k)$ be the category whose objects can be thought of as the affine spaces $\mathbb{A}^n$ over $k$ and whose morphisms $\mathbb{A}^n \to \mathbb{A}^m$ are $m$-tuples of polynomials in $n$ variables over $k$, with composition given by composition of polynomials. On the one hand, this is a Lawvere theory, and the category of commutative algebras over $k$ can be defined as the category of models of it, or more explicitly as the category of product-preserving functors $\text{Poly}(k) \to \text{Set}$. On the other hand, this is a full subcategory of the category of varieties over $k$, and one can try to probe varieties by mapping affine spaces into them; this turns a variety into a presheaf $\text{Poly}(k)^{op} \to \text{Set}$. There is a general relationship between functors and presheaves on the same category called Isbell duality, and the nLab suggests that this is the general setting for adjunctions between things that look like spaces and things that look like commutative algebras, although I haven't really internalized this.
The two points made above can be related as follows. One way to think about the category of sets is that it is the category of ind-objects of the category of finite sets; equivalently, it's the category of presheaves $\text{FinSet}^{op} \to \text{Set}$ sending finite colimits to finite limits. Now, $\text{FinSet}^{op}$ is the category of finite Boolean rings, and the category of Boolean rings can be thought of as the category of ind-objects in this; equivalently, it's the category of functors $\text{FinSet} \to \text{Set}$ preserving finite limits. These two descriptions should be related by Isbell duality.
Why is $\text{FinSet}$ so important, anyway? Well, one categorical property that categories of spaces often share that isn't true in general is that binary products tend to distribute over binary coproducts. This is true in particular in any cartesian closed category, such as $\text{Set}$, and any category with this property behaves in some sense like a categorified commutative ring (!). It turns out that $\text{FinSet}$ is the free distibutive category on a point.