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Every functor $\mathbf{Set} \rightarrow \mathbf{Set}$ I can think of preserves monomorphisms (i.e. injective functions), including:

  • $\mathrm{Hom}(X,-)$,
  • $X \times -$,
  • $X \sqcup -$,
  • and the constant functors.

The monads I can think of all have this property, too.

What are some natural examples that don't?

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  • $\begingroup$ I just found out that in my answer I told you something that you allready knew. See the comment on this answer. $\endgroup$ – drhab Aug 16 '16 at 13:19
  • $\begingroup$ @drhab, the tables have turned! $\endgroup$ – goblin Aug 16 '16 at 13:43
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    $\begingroup$ Of course without the "natural" part there are many answers (e.g. if $F$ is the functor defined on objects by $F(S) = \{1\}$ if $S \neq \emptyset$ and $F(\emptyset) = \{1,2\}$ and on morphisms in the only way possible). $\endgroup$ – Nex Aug 16 '16 at 17:20
  • $\begingroup$ I would have thought any such functor would have to be pretty unnatural, as in @Nex’s example. Interested to see if anyone comes up with a natural example! $\endgroup$ – Robin Houston Aug 16 '16 at 22:06
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    $\begingroup$ Actually, maybe @Nex’s example is more natural than it looks. Isn’t it the same as $\mathrm{Hom}(Hom(-,\varnothing),2)$? $\endgroup$ – Robin Houston Aug 16 '16 at 22:18
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Not a real answer on your question, but it might be enlightening.

Every functor preserves sections (i.e. split monomorphims) and in Set almost all monomorphisms are sections.

The only exceptions are the functions $\varnothing\to Y$ where $Y\neq\varnothing$.

These are the only functions that are monic, but do not have a left-inverse.

So it is not so strange that endofunctors in Set that do not preserve monomorphisms are hard to find.

In order to find a counterexample (or maybe a proof that all monomorphisms are preserved by endofunctors in Set) you will have to focus on the mentioned functions.

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  • $\begingroup$ Your answer implies that one only needs to check if $\emptyset \to \{*\}$ is preserved (since any function $\emptyset \to Y$ where $Y \neq \emptyset$ factors as $\emptyset \to \{*\}\to S$). $\endgroup$ – Nex Aug 16 '16 at 17:15
  • $\begingroup$ @Nex Yes, you are right. Nice observation. $\endgroup$ – drhab Aug 17 '16 at 7:13

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