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We develop Azumaya geometry, which is an extension of classical affinegeometry to the world of Azumaya algebras, and package the informationcontained in all quotient stacks $[\mathrm{rep}_n R\,/\,\mathrm{PGL}_n]$ into apresheaf $\mathrm{Rep}_R$ on it. We show that the classical \'etale and Zariskitopologies extend to Grothendieck topologies on Azumaya geometry in uncountablymany ways, and prove that $\mathrm{Rep}_R$ is a sheaf for all of them. Therestriction to a specific Azumaya algebra $A$ with center $C$ gives us a sheafin 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 $A$.

Azumaya geometry and representation stacks