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I'm studying the concept of faithful functors, but I cannot grasp the difference between being faithtful and being injective on arrows. Could someone explain the difference and provide some examples?

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    $\begingroup$ "Being injective on arrows" is an evil property (not invariant under equivalences of categories). "Faithful" is a good property. $\endgroup$ – Martin Brandenburg Jun 19 '14 at 10:36
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Let's remind ourselves about the definition of a faithful functor: Any functor $$ \mathcal{F} \colon \mathcal{C} \rightarrow \mathcal{D}$$ induces a map on hom-sets $$\mathcal{F}_{X,Y} \colon \mathrm{Hom}_\mathcal{C}(X,Y) \rightarrow \mathrm{Hom}_\mathcal{D}(\mathcal{F}(X),\mathcal{F}(Y))$$ for every pair of objects $X,Y$ in $\mathcal{C}$. Then $\mathcal{F}$ is called faithful if the induced maps $\mathcal{F}_{X,Y}$ are injective for all objects $X,Y$ in $\mathcal{C}$.

The crucial thing to observe is that we are quantifying over objects in the source category $\mathcal{C}$ --- this is what makes the difference between a faithful functor and one injective on morphisms. For instance consider the four-object category $\mathcal{C}$ with $\mathrm{Obj}(\mathcal{C})=\{A,B,C,D\}$ and morphisms given by $$ \mathrm{Hom}(A,B) = \{f_i\}_{i\in\mathbb{Z}}, \\ \mathrm{Hom}(C,D) = \{g_i\}_{i\in\mathbb{Z}}$$ (and for simplicity say the other hom-sets are empty, except for the identity morphisms on each object). If we have another category $\mathcal{D}$ consisting of two objects $P,Q$ and morphisms $$ \mathrm{Hom}(P,Q) = \{h_i\}_{i\in\mathbb{Z}}$$ then the functor $ \mathcal{F} \colon \mathcal{C} \rightarrow \mathcal{D}$ sending $$ A \mapsto P, \\ B \mapsto Q, \\ C \mapsto P, \\ D \mapsto Q $$ and $$ f_i \mapsto h_i, \\ g_i \mapsto h_i$$ (and identity morphisms to identity morphisms, etc.) is faithful, since restricting to every pair of objects $X,Y$ in $\mathcal{C}$ gives an injective map on $\mathrm{Hom}(X,Y)$. On the other hand $\mathcal{F}$ is neither injective on objects or morphisms.

I wanted to emphasize the fact that quantifying over objects in the source category is the important bit: if we required a faithful functor to be injective on the hom-sets between every pair of objects in the target category we would indeed just end up with a functor injective on morphisms.

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A functor that is injective on morphisms is automatically faithful, of course; but by thinking about identity morphisms, one sees that a functor that is injective on morphisms must also be injective on objects.

Exercise. Show that a functor is injective on morphisms if and only if it is injective on objects and faithful.

Once you have proved the claim, you will find many examples. For instance, if $\mathcal{P}$ is a poset considered as a category, then the unique functor $\mathcal{P} \to \mathbb{1}$ is always faithful, hence injective on morphisms if and only if it is injective on objects.

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    $\begingroup$ Oh, I see what you mean. The point is that a functor can identify what were previously different sources and targets. My mistake. $\endgroup$ – Qiaochu Yuan Jun 19 '14 at 6:35
  • $\begingroup$ I guess the analogous result holds with injective replaced by surjective and faithful replaced by full? There any functor $F:\varnothing\to\mathcal{D}$ would serve as corresponding nonexample? $\endgroup$ – C-Star-W-Star Jun 28 '16 at 8:17
  • $\begingroup$ Yes, that's right. $\endgroup$ – Zhen Lin Jun 28 '16 at 9:13

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