Symmetries

TL;DR

This paper explores the idea of incorporating symmetries into set theory by embedding sets into higher categories and groupoids, aiming to provide new explanations for phenomena like mirror symmetry.

Contribution

It proposes a novel approach of embedding sets into higher categories and groupoids to incorporate symmetries, potentially explaining complex mathematical phenomena.

Findings

Sets with symmetries can potentially explain mirror symmetry phenomena.

Embedding sets into higher groupoids offers a new perspective on classical mathematics.

The paper outlines a conceptual framework without empirical validation.

Abstract

Classical mathematics are founded within set theory, but sets don't have \emph{symmetries}. We conjecture that if we allow sets with symmetries, then many problems such as \emph{Mirror symmetry} or \emph{Homological mirror symmetry} can be explained. One way to do this is to embed sets to higher categories and especially into higher groupoids as already envisioned by Grothendieck, and more recently by Voevodsky. We simply outline this idea in these notes.