# On the surjectivity of the map of spectra associated to a tensor-triangulated functor

## TL;DR

This paper investigates the conditions under which a tensor-triangulated functor induces a surjective map on spectra, linking conservativity and nilpotence detection to spectral surjectivity.

## Contribution

It establishes new equivalences between functor properties like conservativity and nilpotence detection and the surjectivity of the induced spectral map.

## Key findings

- F is conservative iff Spc(F) is surjective on closed points
- F detects tensor-nilpotence iff Spc(F) is surjective on the entire spectrum
- Surjectivity of Spc(F) is equivalent to F detecting nilpotence of certain morphisms

## Abstract

We prove a few results about the map $Spc(F)$ induced on tensor-triangular spectra by a tensor-triangulated functor $F$. First, $F$ is conservative if and only if $Spc(F)$ is surjective on closed points. Second, if $F$ detects tensor-nilpotence of morphisms then $Spc(F)$ is surjective on the whole spectrum. In fact, surjectivity of $Spc(F)$ is equivalent to $F$ detecting the nilpotence of some class of morphisms, namely those morphisms which are nilpotent on their cone.