An operad $P = (P(n))$ is an object that encodes algebraic operations $f(x_1, …, x_n)$, of varying arities, and compositions between these operations. The inputs in any algebraic operation of course form a nonempty, finite set. In this talk, we will introduce an abstraction of the concept of an operad, allowing for inputs that come from categories $C$ more general than the category of nonempty, finite sets. We introduce this abstraction in order to better understand the operadic structure on the algebraic varieties that Chen, Gibney, and Krashen introduced in 2006. We will explain how, using a categorical variant of the machinery underlying Fulton-MacPherson compactifications, one can construct an abstract, set-valued operad $FM$ with inputs coming from any given, sufficiently nice category $C$. In certain cases, the operads $FM$ that we construct in this way turn out to be the point sets of scheme-valued operads. We hope to conclude by explaining how these scheme-valued operads subsume Chen-Gibney-Krashen operads, in particular the operad of stable, rational curves, and explaining some of the geometry of the schemes involved.