  We consider representations of quivers taking values in monads or comonads
over a Grothendieck category $\mathcal C$. We treat these as scheme like
objects whose ``structure sheaf'' consists of monads or comonads. By using
systems of adjoint functors between Eilenberg-Moore categories, we obtain a
categorical framework of modules over monad quivers, and of comodules over
comonad quivers. Our main objective is to give conditions for these to be
Grothendieck categories, which play the role of noncommutative spaces. As with
usual ringed spaces, we have to study two kinds of module categories over a
monad quiver. The first behaves like a sheaf of modules over a ringed space.
The second consists of modules that are cartesian, which resemble
quasi-coherent sheaves. We also obtain an extension of the classical
quasi-coherator construction to modules over a monad quiver with values in
Eilenberg-Moore categories. We establish similar results for comodules over a
comonad quiver. One of our key steps is finding a modulus like bound for an
endofunctor $U:\mathcal C\longrightarrow \mathcal C$ in terms of $\kappa(G)$,
where $G$ is a generator for $\mathcal C$ and $\kappa(G)$ is a cardinal such
that $G$ is $\kappa(G)$-presentable. Another feature of our paper is that we
study modules over a monad quiver in two different orientations, which we refer
to as ``cis-modules'' and ``trans-modules.'' We conclude with rational pairings
of a monad quiver with a comonad quiver, which relate comodules over a comonad
quiver to coreflective subcategories of modules over monad quivers.
