Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of $n$-by-$n$ matrices for $n \geq 3$. This obstruction also applies to other spectra such as those named after Zariski, Stone, and Pierce. We extend these no-go results to functors with values in (ringed) topological spaces, (ringed) toposes, schemes, and quantales. The possibility of spectra in other categories is discussed.
Keywords: Ring spectra, Kochen-Specker Theorem
2010 MSC: 16B50, 46L85
Theory and Applications of Categories, Vol. 29, 2014, No. 17, pp 457-474.