Understanding the significance of a functor being full/faithful, especially with adjoints

Question

I'm working through "Basic Category Theory" by Tom Leinster and am trying to get clarity on how to reason about things... one thing I'm not sure about is how to think about what a functor being faithful and/or full gives us (well, beyond the obvious of being injective and/or surjective in the mapping of morphisms and objects).

So later on there is a question: show that for any adjunction, the right adjoint is full and faithful if and only if the counit is an isomorphism.

So I'm trying to think through the "if the right adjoint is full and faithful, the counit is an isomorphism" case. But I'm not really sure how to think about what being full and faithful gives us. Is it that we know there is a section (axiom of choice!!) and a retract? Is it that there has to be an inverse? (I don't think this is true, right? just because there are separate morphisms that will cancel it out on either side doesn't mean there is one that does both...right?)

So I'm at a loss of how to use full/faithfulness to reason about whether there is an isomorphism, because those properties only related to the right adjoint, not the left.

I'm probably missing something obvious. Thanks for your help.

Update: am trying to do research on this and haven't found a proof, but have found an alternate statement here: http://ncatlab.org/nlab/show/reflective+subcategory. See proposition one.

Answer 1

I can't look at the reference you mentionwd right now, but the theorem you mention (and more) is proved in Maclane' CWM, 2nd Ed page 90. It is not exactly trivial, expecially for a beginner, MacLane needs a lemma and takes half a page to prove it. The whole story is this:

• the right adjoint is faithful iff the components of the counit are epi
• the right adjoint is full iff the components of the counit are split mono

Your theorem is a consequence of the 2 propositions above.

Regarding the "significance" of full and faithful, they are just a way to express the injectivity and surjectivity of the mapping of Homs to Homs. You could think of them as "local" properties, in the same sense that "locally small" means that the Homs are small (but not necessarily the whole category). How do you use these properties? Well exactly like you would in normal set theory (the Homs are sets). Just unwrap them.

Answer 2

You can think of a functor $F : C \to D$ being faithful as roughly meaning that the objects of $C$ can be thought of as objects in $D$ with extra structure, the idea being that writing down a morphism between two objects in $C$ is the same as writing down a morphism between their images in $D$ satisfying some extra property, to be interpreted as meaning it preserves some extra structure. For example, the forgetful functor $\text{Grp} \to \text{Set}$ is faithful.

You can think of a functor $F$ being both full and faithful as roughly meaning that the objects of $C$ can be thought of as objects in $D$ satisfying a property, the idea being that being fully faithful means that writing down a morphism between two objects in $C$ is the same as writing down a morphism between their images in $D$. For example, the inclusion functor $\text{Ab} \to \text{Grp}$ is both full and faithful.

See also the nLab. It's unclear to me whether there's any good conceptual picture of what it means for a functor to be full but not faithful.