Exploring mathematical objects from custom-tailored mathematical universes

TL;DR

This paper explores various mathematical universes called toposes, highlighting their unique properties and how they offer different perspectives on standard mathematical objects, revealing the diversity beyond the usual framework.

Contribution

It introduces specific alternative toposes and examines their peculiar properties, illustrating how toposes serve as diverse mathematical universes with distinct logical and structural features.

Findings

Identification of toposes where the axiom of choice fails

Examples of toposes where the intermediate value theorem does not hold

Demonstration of how toposes provide different lenses for viewing mathematical objects

Abstract

Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are toposes in which the axiom of choice and the intermediate value theorem from undergraduate calculus fail. The purpose of this contribution is to give a glimpse of the toposophic landscape, presenting several specific toposes and exploring their peculiar properties, and to explicate how toposes provide distinct lenses through which the usual mathematical objects of the standard topos can be viewed.