Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W\_{dom}:A\to W$ and $i^W\_{cod}:B\to W$ to denote the boundary-component inclusions. Let $\pi\_0:{\bf Man}\to{\bf Set}$ denote the connected components functor.


Let $\mathcal{L}\subset Mor\_{1-Cob}$ denote the set of morphisms $W$ for which the function $\pi\_0(i^W\_{cod}\,):\pi\_0(B)\to \pi\_0(W)$ is an injection and for which the 1-manifold $W$ contains no closed loops, $\not\exists (S^1\hookrightarrow W)$.


Let $\mathcal{R}\subset Mor\_{1-Cob}$ denote the set of morphisms $W$ for which the function $\pi\_0(i^W\_{dom}):\pi\_0(A)\to \pi\_0(W)$ is an injection. 


I believe that $\mathcal{L}$ and $\mathcal{R}$ form an orthogonal factorization system on $1-Cob$, but I don't know how to prove it. 


**Question**: Can you find a proof, a reference, or a counterexample for this conjecture?


  


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(Just for symmetry....)


Let $\mathcal{L'}\subset Mor\_{1-Cob}$ denote the set of morphisms $W$ for which $\pi\_0(i^W\_{cod})$ is an injection.


Let $\mathcal{R'}\subset Mor\_{1-Cob}$ denote the set of morphisms $W$ for which $\pi\_0(i^W\_{dom})$ is an injection and for which the 1-manifold $W$ contains no closed loops, $\not\exists (S^1\hookrightarrow W)$. 


I also believe that $\mathcal{L'}$ and $\mathcal{R'}$ form an orthogonal factorization system on $1-Cob$, but I don't know how to prove it either.




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Edit provenance: In an earlier version of this question, I had reversed $\mathcal{L}$ and $\mathcal{R}$. This allowed Chris Schommer-Pries to correctly answer my question as originally posed, but not the question I meant to ask (i.e., the one you see above now). 



An orthogonal factorization system on 1-Cob?
This question was answered in the affirmative by Joseph Abadi, [here](http://arxiv.org/pdf/1506.03119v1.pdf).


