Spectra as universal objects in categories of supports

Many years ago, André Joyal outlined a method of describing the Zariski spectrum $Spec(R)$ of a commutative ring $R$ in a manner that makes no reference to prime ideals of $R$. In Joyal’s approach, the spectrum is not a topological space, but a distributive lattice that satisfies a certain universal property. Recently, this approach has been shown to be very fruitful in understanding other spectra, such as the spectrum of a tensor triangulated category. In this paper, we take a similar method to describe as universal objects several other ‘spectrum like spaces’ that arise in commutative algebra and noncommutative algebraic geometry.