A monomorphism and epimorphism that is neither injective nor surjective(set-theoretically)

Question

Is it possible to construct a category such that there is a monomorphism and epimorphism that is neither injective nor surjective(set-theoretically).

I notice that there is a solution in which a monomorphism and epimorphism is not surjective. I wonder if a morphism always has to be "injective" as long as it is a monomorphism and epimorphism.

Answer 1

Consider this silly category:

Object: Sets $A$ such that $\{0,1\}\subseteq A\subseteq \mathbb N$.

Morphisms: Maps $f:A\to B$ with the properties $$ f(0)=f(1)=0 \\ x\notin\{0,1\}\Rightarrow f(x)\notin\{0,1\} $$

Composition: Function composition.

Identity: $\mathrm{id}(x) = \begin{cases} 0 &\text{if }x=1 \\ x & \text{otherwise} \end{cases}$

Then no morphism at all is either injective or surjective as a map -- but there are plenty of epimorphisms and monomorphisms, such as all the identities.

Answer 2

Yes, for instance, define a subcategory of $\mathbf{Set}$ whose only objects are $$A=\{1,2\}\qquad\text{and}\qquad B=\{3,4\}$$ and morphisms are given as $$\begin{align}\operatorname{Hom}(A,A)&=\{\operatorname{id}_A\},\\\operatorname{Hom}(A,B)&=\{f\},\\\operatorname{Hom}(B,A)&=\emptyset,\\\operatorname{Hom}(B,B)&=\{\operatorname{id}_B\},\end{align}$$ where $f:A\to B$ is defined by $f(1)=f(2)=3$. Then $f$ is a monomorphism and an epimorphism, but it is neither injective nor surjective.