Definable functoriality of tensor-triangular spectra
Isaac Bird, Jordan Williamson
TL;DR
The paper addresses functoriality of tensor-triangular spectra under definable functors between rigid tt-categories, reframing the phenomenon through purity and the definable-pure correspondence. It constructs a continuous map on the homological spectrum $Spc^h$ using $FJ_B$ and the kernel of $-\otimes FJ_B$, and shows this induces a Balmer-spectrum map via the Kolmogorov quotient. The main theorem requires $F$ to have a left adjoint $\Lambda$ with a projection formula, and it recovers Balmer's functoriality for geometric functors as a special case. The approach provides a purity-centered, definable-framework for functoriality that may extend beyond rigid tt-categories and offers a new perspective on how purity governs spectrum-level behavior.
Abstract
We prove that the homological and Balmer spectra in tensor-triangular geometry are functorial in certain definable functors, thereby providing an alternative perspective on functoriality in tensor-triangular geometry from the viewpoint of purity, and generalising current results in the literature.