Differences

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--- monomorphism [2015/03/16 23:44]
nikolaj
+++ monomorphism [2015/03/16 23:48]
nikolaj
@@ Line 23: / Line 23: @@
 Say $f$ is not an injection and hence collapses two different terms $a,d\in A$ into a single value, i.e. $f(a)=f(d)$. The pullback object then contains $\langle a,a\rangle$ and $\langle d,d\rangle$, but also $\langle a,d\rangle$ and $\langle d,a\rangle$. So $A\times_BA$ is bigger than $A$. On the other hand, if $f$ is an injection, then for any $a$, the pullback only contains $\langle a,a\rangle$. We get $A\times_BA\cong A$.
-The definition "$\langle A,\prod_{A}1_A\rangle$ is a pullback of $f$ along itself" implies that $A$ is a valid pullback already, because $f$ doesn't collapse any information.
+The definition "$\langle A,\prod_{A}1_A\rangle$ is a pullback of $f$ along itself" implies that $A$ is already a valid pullback object, because $f$ doesn't collapse any information. Dually, the definition of an epimorphism (surjections in ${\bf{Set}}$) implies that $A$ is a valid pushout $A+_BA$, an object which generally contains less information than $A$.
 A stepwise characterization of a monomorphism from the pullback, with a focus on the universal property, is given in [[pullback . category theory|pullback]].