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What happens if we define a functor $F:C \to D$ to be injective when it is injective on isomorphism classes, or equivalently when it gives an injection from the objects of the skeleton of $C$ to the skeleton of $D$?

Edit: To be more specific, how does this definition relate to a that of a fully faithful functor?

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    $\begingroup$ Although I regularly ask myself questions of the form "What happens if I make this definition / construction / etc.", I think this question needs to be more specific. Do you want to compare this notion to other "injective-like" conditions on functors or something? $\endgroup$
    – Tim Campion
    Feb 11, 2021 at 16:19
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    $\begingroup$ Your definition is the notion of "essentially injective functor", i.e. $F(A) \cong F(B) \implies A \cong B$, which is the analogue of essentially surjective functor for injective functions. $\endgroup$
    – varkor
    Feb 11, 2021 at 16:30
  • $\begingroup$ @Varkor: How does this relate to fullness and faithfullness? $\endgroup$ Feb 11, 2021 at 16:33

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Call $F : \mathbf C \to \mathbf D$ essentially injective (mirroring the definition of essentially surjective functor) if $F(A) \cong F(B)$ implies that $A \cong B$. This matches your definition: a functor between skeletal categories is essentially injective if and only if its object function is injective.

Recall that a functor is conservative if, whenever $F(g) : F(A) \to F(B)$ is an isomorphism, $g : A \to B$ is also an isomorphism.

These two notions are related: in particular, a conservative functor that is full is also essentially injective, because if $F(g)$ is an isomorphism, then so is $g$ and hence $A \cong B$. Fullness ensures that every morphism between $F(A)$ and $F(B)$ is in the image of $F$, so if an isomorphism exists, it will necessarily be reflected. (As Mike Shulman points out in the comments, it suffices for $F$ to be pseudomonic, i.e. full on isomorphisms and faithful.)

Finally, every fully faithful functor is conservative, and hence essentially injective. It is easy to see that the converse does not hold: for instance, any functor from the category with two objects and a single isomorphism is essentially injective, but will rarely be fully faithful.

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    $\begingroup$ A weaker notion than fully-faithful which still implies conservativity and essential injectivity is being pseudomonic. $\endgroup$ Feb 11, 2021 at 18:57

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