Idempotents in Tangle Categories Split

TL;DR

This paper explores the structure of tangle categories using 3-manifold techniques, demonstrating that every idempotent morphism splits into a composition involving an identity, revealing new structural insights.

Contribution

It introduces a novel approach to analyze tangle categories by applying 3-manifold techniques to show idempotent morphisms split in a natural way.

Findings

Every idempotent morphism splits as A=B∘C

C∘B is an identity morphism

Provides new structural understanding of tangle categories

Abstract

In this paper we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism $A$ in such a category naturally splits as $A=B\circ C$ such that $C\circ B$ is an identity morphism.