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In this question it was asked what relation, if any, exists between the notions of a monomorphism in a general category $\mathcal C$ and an injective map. Of course the question only makes sense if $\mathcal C$ is a concrete concretizable category. If $\pi:\mathcal C\to Set$ is a faithful functor, we may regard a morphism $f:X\to Y$ of objects of $\mathcal C$ as an injection (relative to $\pi$) if $\pi(f)$ is injective in $Set$.

One then notices that this notion depends very much on the choice of concretization, and thus that this notion is not an intrinsic property of the category itself.

However, Emil Jerabek noticed that it is natural to ask the following: given a concrete concretizable category, does it admit a concretization which preserves monomorphisms?

Apologies if this question is trivial, hopelessly too general, or of no interest to category theorists.

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In this question it was asked what relation, if any, exists between the notions of a monomorphism in a general category $\mathcal C$ and an injective map. Of course the question only makes sense if $\mathcal C$ is a concrete category. If $\pi:\mathcal C\to Set$ is a faithful functor, we may regard a morphism $f:X\to Y$ of objects of $\mathcal C$ as an injection (relative to $\pi$) if $\pi(f)$ is injective in $Set$.

One then notices that this notion depends very much on the choice of concretization, and thus that this notion is not an intrinsic property of the category itself.

However, Emil Jerabek noticed that it is natural to ask the following: given a concrete category, does it admit a concretization which preserves monomorphisms?

Apologies if this question is trivial, hopelessly too general, or of no interest to category theorists.

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# Existence of "good" concretizations

In this question it was asked what relation, if any, exists between the notions of a monomorphism in a general category $\mathcal C$ and an injective map. Of course the question only makes sense if $\mathcal C$ is a concrete category. If $\pi:\mathcal C\to Set$ is a faithful functor, we may regard a morphism $f:X\to Y$ of objects of $\mathcal C$ as an injection (relative to $\pi$) if $\pi(f)$ is injective in $Set$.

One then notices that this notion depends very much on the choice of concretization, and thus that this notion is not an intrinsic property of the category itself.

However, Emil Jerabek noticed that it is natural to ask the following: given a concrete category, does it admit a concretization which preserves monomorphisms?

Apologies if this question is trivial, hopelessly too general, or of no interest to category theorists.