  We develop Azumaya geometry, which is an extension of classical affine
geometry to the world of Azumaya algebras, and package the information
contained in all quotient stacks $[\mathrm{rep}_n R\,/\,\mathrm{PGL}_n]$ into a
presheaf $\mathrm{Rep}_R$ on it. We show that the classical \'etale and Zariski
topologies extend to Grothendieck topologies on Azumaya geometry in uncountably
many ways, and prove that $\mathrm{Rep}_R$ is a sheaf for all of them. The
restriction to a specific Azumaya algebra $A$ with center $C$ gives us a sheaf
in the \'etale topology which is represented by an affine $C$-scheme
$\mathrm{rep}_A(R)$, which we call the Azumaya representation scheme of $R$
with respect to $A$.
