On the algebraic characterization of the category of 3-dimensional cobordisms

TL;DR

This paper establishes a functorial relationship between two algebraic categories generated by Hopf algebra objects, providing an alternative set of axioms for the algebraic characterization of the 3-dimensional cobordism category.

Contribution

It constructs a functor from the algebraic category alg to the universal algebraic category alh, offering a new perspective and axiomatization for the algebraic description of 3D cobordisms.

Findings

Established a functor algalh

Provided an alternative axiomatization for alh

Linked two algebraic models of 3D cobordisms

Abstract

It is proved in \cite{BP} (arXiv:1108.2717) that the category of relative 3-dimensional cobordisms $\cal Cob^{2+1}$ is equivalent to the universal algebraic category $\overline{\overline{\cal H}}{}^r$ generated by a Hopf algebra object. A different algebraic category $\overline{\cal Alg}$ generated by a Hopf algebra object is defined in \cite{AS} and it conjectured to be equivalent to $\cal Cob^{2+1}$ as well. We prove that there exists a functor $\overline{\cal Alg}\to \overline{\overline{\cal H}}{}^r$, and use it to present an alternative set of axioms for $\overline{\overline{\cal H}}{}^r$.