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I am required to find such an example in Exercise 3.e. in Topology: A Categorical Approach. I considered the category $\textsf{Top}$, and proved that if taken for granted that every monomorphism in $\textsf{Top}$ is injective, there wouldn't be such an example in the category. However, I am neither able to prove the granted assertion nor to find such an example in a familiar category, rather in a technical and constructed one, and help is needed regarding these two questions.

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SalutaFungo
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1 Answers1

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3 points to be made:

  1. I was right that a morphism in $\textsf{Top}$ is monic iff it is an injective continuous function.

  2. The question referred to in Chris Eagle's comment indeed provides a counterexample that despite 1, there can exist monomorphisms in both directions despite two spaces are not homeomorphic. In my incorrect proof, I falsely thought that isomorphism between underlying set $\iff$ same underlying set.

  3. In $\textsf{Grp}$, there exists an embedding $F_A$ into $F_{\{a,b\}}$, where $A$ is any countable set with cardinality bigger than $2$, and $F_X$ is the free group generated by the set $X$. Together with the obvious injection $F_{\{a,b\}} \to F_A$, this is another example.

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