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?

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