TITLE: Can an algebraic variety be described as a category, in the same way as a group?
QUESTION [12 upvotes]: Can an algebraic variety be described as a category, in the same way as a group? A group can be considered a category with one object, with elements of the group the morphisms on the object.

REPLY [10 votes]: The conjectural remarks by Qiaochu are supported by the following theorem (joint work with Alexandru Chirvasitu):

Let $X,Y$ be schemes, where $X$ is quasi-compact and quasi-separated. Then $f \mapsto f^*$ establishes an equivalence between $\hom(Y,X)$ and the category of cocontinuous symmetric monoidal functors $\mathsf{Qcoh}(X) \to \mathsf{Qcoh}(Y)$.

Therefore, the category of (nice) schemes embeds fully faithfully into the $2$-category of cocomplete symmetric monoidal categories. Large parts of algebraic geometry can be translated and generalized by means of this embedding; this will be hopefully explained in full detail in my thesis. For example, under suitable finiteness assumptions, one can construct the "projective tensor bundle" $\mathbb{P}_{\otimes}(\mathcal{E})$ for an object $\mathcal{E}$ of a cocomplete symmetric monoidal category $\mathcal{C}$, which classifies invertible quotients of $\mathcal{E}$, and when $\mathcal{C}=\mathsf{Qcoh}(S)$ for some scheme $S$ this coincides with $\mathsf{Qcoh}(\mathbb{P}_S(\mathcal{E}))$.
In this picture, the monoidal structure is crucial; there are lots of auto-equivalences of $\mathsf{Qcoh}(X)$ which are not induced by automorphisms of $X$. On the other hand, if $X,Y$ are quasi-separated schemes such that $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ are equivalent as abstract categories, then $X$ and $Y$ are isomorphic; this is a quite deep theorem (proved by Gabriel for noetherien schemes, then generalized by Rosenberg, and corrected by Gabber) and has motivated a new perspective on non-commutative algebraic geometry using abelian categories (see the work by Artin, Zhang, Rosenberg and others).