  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 nxn-matrices for n at least 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.
