On the Existence of an Orthogonal Factorization System on 1-Cob and 2-Cob

TL;DR

This paper establishes the existence of orthogonal factorization systems on the categories 2-Cob and oriented 1-Cob by combinatorial definitions and functorial relationships.

Contribution

It introduces a combinatorial definition of 2-Cob and demonstrates the existence of orthogonal factorization systems on both 2-Cob and oriented 1-Cob using functorial methods.

Findings

Orthogonal factorization system on 2-Cob proven

Functor from oriented 1-Cob to 2-Cob constructed

Orthogonal factorization system on oriented 1-Cob established

Abstract

We define the category 2-Cob combinatorially and use this definition to prove the existence of an orthogonal factorization system. In the second half of the paper, we define oriented 1-Cob similarly and define a functor from oriented 1-Cob to 2-Cob. After defining this functor, the orthogonal factorization system on 2-Cob is used, in turn, to prove the existence of an orthogonal factorization system on oriented 1-Cob.