Examples of faithful but not-full functors between two categories in both directions?

Question

A functor F:C→D from a category C to a category D is said to be full if, for each pair of objects, x,y ∈ C, the function,

F:C(x,y)→D(F(x),F(y))

between hom sets is surjective.

A functor F:C→D from a category C to a category D is said to be faithful if for each pair of objects x,y∈C, the function

F:C(x,y)→D(F(x),F(y)) between hom sets is injective.

What I am unsure about is, for categories C and D, can you have a functor that is faithful but not full in both directions? That is, a category F:C→D and G:D→C, such that both F and G are faithful but not full.

In particular, are there any simple examples of this?

Answer 1

Lots of examples. Here’s two.

1. Take $C=D=\mathbf{Set}$, and $F=G$ the direct image functor, taking a set $X$ to its power set $P(X)$, and taking a function $f\colon X\to Y$ to the direct image function $\overline{f}\colon P(X)\to P(Y)$ sending $A\subseteq X$ to $f(A)=\{f(a)\mid a\in A\}\subseteq Y$. This is faithful: if $f\neq g$, then there exists $x\in X$ such that $f(x)\neq g(x)$; then $\overline{f}(\{x\})=\{f(x)\}\neq\{g(x)\}=\overline{g}(\{x\})$. It is not full, because the direct image functions always take singletons to singletons, but there are certainly functions $P(X)\to P(Y)$ (for $X$ and $Y$ nonempty) that take singletons in $P(X)$ to things that are not singletons in $P(Y)$.

2. For an example with $C\neq D$, let $C=\mathbf{Set}$, $D=\mathbf{Group}$, let $F\colon C\to D$ be the free group functor (taking $X$ to the free group on $X$), and let $G\colon D\to C$ be the underlying set functor, taking a group to its underlying set. Then both $F$ and $G$ are faithful but not full. This can be generalized by replacing $D$ with the category of all rings with unity, semigroups, abelian groups, and vector spaces, among others.

Answer 2

As Arturo said, there are a lot of example, but we even think we can go as far as saying that this question is probably more a set-theoretic one than a categorical one. I will construct a small example to illustrate my claim:

Consider the category $\mathcal{C}$, which has two objects $A$ and $B$, and such that $\mathcal{C}(A,B) = \mathbb{Z}$. Note that there is a function $\mathbb{Z}\to\mathbb{Z}$ that is injective but not surjective (take for instance $f(k)=2k$). Then from this function, you can construct a functor $F:\mathcal{C}\to\mathcal{C}$ which is faithful but not full: just send $A$ and $B$ onto themselves, and act on the morphisms as $f$. Since $F$ goes from $\mathcal{C}$ to itself, you can also chose $F$ for the functor in the other direction.

What is at play here is that the categories with $2$ objects really are the same as sets, and the question you are asking, translated in this context really boils down to "is there a pair of injective, but non surjective function?"

As a side note, your question got me thinking whether there is a Cantor-Bernstein theorem for categories, and that sounds pretty interesting and I have never thought about it.