I hope this question isn't too silly. It is certainly fundamental, so the answer is likely contained at least implicitly in most sources out there, but I haven't seen it done this way (that is, in this particular functorial manner) in a way which is overt enough for me to catch on. I am familiar with the classical $Proj$-construction for a graded ring, so that's not quite what I'm looking for.
Let $k$ be a ring. Let's call a covariant functor of sets on some category of $k$-algebras an algebraic functor (over $k$). The affine $I$-space over $k$ is the algebraic functor $\mathbb{A}^I:(k−alg)→(set)$ which takes a $k$-algebra $R$ to the set $\mathbb{R}^I$ of $I$-tuples of elements of $R$. This functor is (co)representable by the ring $k[T_i],i∈I$, so $\mathbb{A}^I$ is (represented by) an affine scheme.
I want the projective space over $k$ in terms of an algebraic functor over $k$. I'm thinking something like $R↦\{\mathbb{R}^{I+∞}/\mathbb{G}_m(R)\}$ (where $\mathbb{G}_m(R)$ is the multiplicative group of $R$), or as a functor sending $R$ to some set of modules of rank $1$. One should then be able to show that it has a cover by four copies of the affine $I$-space over $k$. Alternatively, it would likely make sense to consider it a functor on the category of graded $k$-algebras.