Recall that $\operatorname{Sch}$ can be identified with the subcategory of (Zariski-)locally representable (by an affine) étale sheaves on $\operatorname{CRing^{op}}$. In this case, $\operatorname{Spec}(-):\operatorname{CRing^{op}}\to \operatorname{Sch}$ is simply the co-Yoneda embedding $R\mapsto \operatorname{Hom}(R,-)$ (which makes sense since the étale topology is subcanonical and $\operatorname{Hom}(R,-)$ is obviously locally affine).
Is there a nice "sheaf-theoretic" description of $\operatorname{Proj}:\operatorname{(Gr_{{\mathbf Z}_{\geq 0}}CRing)^{op}}\to \operatorname{Sch}$ (I've only seen $\operatorname{Proj}$ for nonnegative integral grading. If we can use more exotic gradings, I guess I'd be interested in that too). Hoping for it to be as nice as $\operatorname{Spec}$ seems like a bit of a pipe dream, but I'm wondering if there is a nicer way to describe it than Hartshorne's approach, which feels rather arbitrary.
Edit: To clarify, I'm looking for a construction $\operatorname{Proj}:\operatorname{(Gr_{{\mathbf Z}_{\geq 0}}CRing)^{op}}\to \operatorname{Sh}_{\acute{et}}(\operatorname{CRing^{op}})$, which we can then see lands in $\operatorname{Sch}$ by showing that we can cover it with Zariski-open affines.
Best Answer
Perhaps I've not understanding your question, but it sounds like you're asking "What is the functor of points of a Proj?"
The answer, of course, is the functor that sends a scheme $X$ to the set of line bundles $L$ on that scheme equipped with a graded map of $R$ to $\Gamma(X;\oplus_{n\geq 0} L^n)$ whose image generates $L^n$ as a $\mathcal{O}_X$-module. The affine open sets are given by the subset where a positive degree element $r$ of $R$ is sent to non-vanishing section; these open sets are easily seen to be the Spec of the degree 0 part of $R_r$, the localization of $R$ at $r$.
EDIT: Let me incorporate some of the things I said below: if you want to just work with rings, then replace "line bundle" with "invertible projective" and $\Gamma(X;\oplus_{n\geq 0} L^n)$ with "tensor algebra over A."
If you're OK with sheafifying, you can do something simpler, which is assume that your line bundle is trivial (but not with a fixed trivialization), i.e. that your invertible projective is free of rank 1 (but not canonically isomorphic to A).
A graded map from $R$ to the tensor algebra of a rank 1 free module is the same as a map from $R\to A$, after you pick an isomorphism of that module to $A$. However, you have to identify maps that come from the same map to the tensor algebra under different isomorphisms (i.e. mod out by the action of the units of A), and throw out maps where the images of the degree 1 elements don't generate A (this is why you get a Proj-ish thing).