Decomposition-space slices are toposes

TL;DR

This paper demonstrates that the slice category over any decomposition space forms a presheaf topos, revealing a deep structural property of decomposition spaces and CULF maps.

Contribution

It establishes that the slice category of decomposition spaces over any given space is a presheaf topos, connecting decomposition spaces to topos theory.

Findings

Slice category over any decomposition space is a presheaf topos.

The category of decomposition spaces and CULF maps is locally a topos.

Provides a new perspective on the structure of decomposition spaces.

Abstract

We show that the category of decomposition spaces and CULF maps is locally a topos. Precisely, the slice category over any decomposition space D is a presheaf topos, namely decomp/D=Psh(tw D).