# pullbacks of epi-mono factorizations

Let $\mathcal{C}$ be a category with the following three properties:

1) $\mathcal{C}$ is finitely complete

2) $\mathcal{C}$ has an extremal-epi-mono factorization

3) extremal epimorphisms are stable under pullbacks

Now, consider a morphism $f:A\to B$ in $\mathcal{C}$ and a pullback of $f$ along an arbitrary arrow $g$:

$$\begin{array} PP & \stackrel{f'}{\longrightarrow} & C \\ \downarrow{g'} & & \downarrow{g} \\ A & \stackrel{f}{\longrightarrow} & B \end{array}$$

By property 2), we can write $f$ as $f=m\circ e$, with $e$ an extremal epimorphism and $m$ a monomorphism. The same can be told about $f'$, so $f'=m'\circ e'$. Now consider the diagram $$\begin{array} PP & \stackrel{e'}{\longrightarrow} & I' & \stackrel{m'}{\longrightarrow}&C \\ \downarrow{g'} & & \downarrow{g''} & &\downarrow{g} \\ A & \stackrel{e}{\longrightarrow} & I & \stackrel{m}{\longrightarrow}&B \end{array}$$ Is it true that $(P,e',g')$ is a pullback of $(e,g'')$ and that $(I',m',g'')$ is a pullback of $(m,g)$, for a suitable $g''$?

In other words, is it true that not only extremal epimorphisms, but the whole epi-mono factorization is stable under pullbacks?

• Yes, under the above assumptions, extremal-epi-mono factorisations are stable under pullbacks. To prove stablity under pullbacks, you first define $(I',m',g'')$ to be the pullback of $(m,g)$ and $(P,e',g')$ to be the pullback of $(e,g'')$. Then, notice that $(P, m'\circ e',g')$ is the pullback of $(m \circ e,g)=(f,g)$. – user337830 Nov 17 '16 at 16:32