Definable functoriality of tensor-triangular spectra

TL;DR

This paper demonstrates that the homological and Balmer spectra in tensor-triangular geometry are functorial under certain definable functors, offering a new perspective through purity and extending existing results.

Contribution

It establishes the functoriality of spectra in tensor-triangular geometry via definable functors, broadening the theoretical framework and generalizing prior findings.

Findings

Spectra are functorial under certain definable functors.

Provides an alternative perspective based on purity.

Generalizes previous results in the literature.

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.