Free factorizations (Q5957801)

Let \(\mathcal K\) be a category. We say that a functor \(F:\mathcal K \to\mathcal L\) preserves a class \(\mathcal E\) of \(\mathcal K\)-epimorphisms (or a class \( \mathcal M\) of \(\mathcal K\)-monomorphisms) if \(F(\mathcal E)\) (or \(F(\mathcal M)\)) is a subclass of \(\mathcal L\)-epimorphisms (or \(\mathcal L\)-monomorphisms). Let \(f\) be a \(\mathcal K\)-morphism. The following statements are proved: 1) If \((\mathcal E,\mathcal M)\) is an orthogonal pair in \(\mathcal K\) then there exists an embedding \(E^1:\mathcal K \to\mathcal K_1\) preserving \(\mathcal E\) and \(\mathcal M\) such that \( E^1f\) admits a (retraction-section)-factorization \(E^1f=r\circ s\) and \(E^1\), \(r\) and \(s\) are universal with this property. Precisely, if \(F:\mathcal K \to \mathcal L\) is a functor preserving \(\mathcal E\) and \(\mathcal M\) and if \( Ff\) admits a (retraction-section)-factorization \(Ff=\bar {r}\circ\bar { s}\) then there exists a unique functor \(F^{\#}:\mathcal K_1 \to \mathcal L\) with \(F=F^{\#}\circ E^1\), \(F^{\#}r=\bar {r}\) and \(F^{\#}s=\bar {s}\). Then \( (E^1(\mathcal E),E^1(\mathcal M))\) is an orthogonal pair. 2) If \(\mathcal E\) is a class of \(\mathcal K\)-epimorphisms then there exists an embedding \(E^2:\mathcal K \to \mathcal K_2\) preserving \(\mathcal E\) such that \( E^2f\) admits an (epi-section)-factorization \(E^2f=r\circ e\) and \(E^2\) and \(r\) are universal with this property. Then \(E^2\) preserves monosources and orthogonal pairs. 3) There exists an embedding \(E^3:\mathcal K \to\mathcal K_3\) such that \( E^3f\) admits an (epi-mono)-factorization \(E^3f=m\circ e\) and \(E^3\) and \(m\) are universal with this property. Then \(E^3\) is full and preserves monosources and episinks. \ Several consequences of these results are derived.