# Do pushouts preserve regular monomorphisms?

In $\mathbf{Top}$, if $A$ is a subspace of $X$ and $f: A \to Y$ is a continuous mapping, then $Y$ embeds into $X \cup_f Y$. I wonder if this generalizes to an arbitrary category.

Consider the pushout of a cospan $Y \leftarrow A \hookrightarrow X$, where the latter arrow is a regular monomorphism. Is the arrow parallel to the regular monomorphism itself a regular monomorphism?

• $X \cup_f Y$ is called a pushout, not a pushforward. – Zhen Lin Dec 2 '12 at 14:35
• @ZhenLin OMG, I spent two years reading it as "pushforward" :( – Alexei Averchenko Dec 2 '12 at 14:41
• After dualizing your question is equivalent to: "is the pullback of a regular epimorphism a regular epimorphism?". The answer is positive for regular categories. And this make me suspect that we can construct a counterexample for the general situation. However, I failed in finding one, for the moment. – Mauro Porta Dec 4 '12 at 21:59
• Thanx for your question. I wanted to ask it myself, because just today I noticed that monos are preserved by pushouts in any category of modules over a ring. Btw. I used this property to prove that modules with the property -if submodule then direct summand- are actually injectives. I don't know, whether this is a coincidence. – bbxlmnistvii Jan 25 '15 at 15:36

The claim fails in $\textbf{Top}^\textrm{op}$, amusingly. Let $A = \{ a, b, c, d \}$ with open sets $$\emptyset, \{ a \}, \{ b \}, \{ a, b \}, \{ a, c \}, \{ b, d \}, \{ a, b, c \}, \{ a, b ,d \}, \{ a, b, c, d \}$$ let $B = \{ 0, 1, 2 \}$ with open sets $$\emptyset, \{ 0 \}, \{ 0, 1 \}, \{ 0, 1, 2 \}$$ and let $C = \{ 0, 2 \}$ be topologised as a subspace of $B$. Let $p : A \to B$ be the continuous map given by $$p(a) = 0, p(b) = 1, p(c) = 1, p(d) = 2$$ and let $i : C \to B$ be the inclusion. It is not hard to see that $p$ is a quotient map, so it is a regular epimorphism; now consider the pullback of $p$ along $i$. This is the map $q : D \to C$ where $D = \{ a, d \}$ is a discrete subspace of $A$, and $q$ is certainly surjective (so is an epimorphism) but $q$ is not a quotient map (so not a regular epimorphism). Thus, pullbacks in $\textbf{Top}$ do not preserve regular epimorphisms; equivalently, pushouts in $\textbf{Top}^\textrm{op}$ do not preserve regular monomorphisms.